Majorana zero modes in iron-based superconductors

The Majorana zero mode (MZM), which manifests as an exotic neutral excitation in superconductors, is the building block of topological quantum computing. It has recently been found in the vortices of several iron-based superconductors as a zero-bias conductance peak in tunneling spectroscopy. In particular, a clean and robust MZM has been observed in the cores of free vortices in (Li0.84Fe0.16)OHFeSe. Here using scanning tunneling spectroscopy, we demonstrate that Majorana-induced resonant Andreev reflection occurs between the STM tip and this zero-bias bound state, and consequently, the conductance at zero bias is quantized as 2e2/h. Our results present a hallmark signature of the MZM in the vortex of an intrinsic topological superconductor, together with its intriguing behavior.

We classify gapped topological superconducting (TSC) phases of one-dimensional quantum wires with local magnetic symmetries, in which the time-reversal symmetry }{}mathcal {T} is broken, but its combinations with certain crystalline symmetries, such as }{}M_x mathcal {T}, }{}C_{2z} mathcal {T}, }{}C_{4z}mathcal {T} and }{}C_{6z}mathcal {T}, are preserved. Our results demonstrate that an equivalent BDI class TSC can be realized in the }{}M_x mathcal {T} or }{}C_{2z} mathcal {T} superconducting wire, which is characterized by a chiral Zc invariant. More interestingly, we also find two types of totally new TSC phases in the }{}C_{4z}mathcal {T} and }{}C_{6z}mathcal {T} superinducting wires, which are beyond the known AZ class, and are characterized by a helical Zh invariant and Zh⊕Zc invariants, respectively. In the Zh TSC phase, Z pairs of Majorana zero modes (MZMs) are protected at each end. In the }{}C_{6z}mathcal {T} case, the MZMs can be either chiral or helical, and even helical-chiral coexisting. The minimal models preserving }{}C_{4z}mathcal {T} or }{}C_{6z}mathcal {T} symmetry are presented to illustrate their novel TSC properties and MZMs.

Spin Seebeck effect induced by a Majorana zero mode in a nanomagnet

Majorana zero modes (MZMs) at the boundary of topological superconductors (TSC) have garnered significant attention for their application in quantum computation. In this study, we demonstrate that in heterostructures composed of the one-dimensional (1D) chiral nanowire and a superconductor, structural chirality could induce the spin-triplet-polarized MZMs. Interestingly, the opposite spin-polarized MZMs are robustly localized at the end of nanowire, whose direction depends on the chirality. Furthermore, this spin-triplet polarization gives rise to chirality-dependent spin-polarized local Andreev reflection (LAR) processes at the TSC/normal metal interface, providing a mechanism for detecting MZMs in TSC-based devices. Finally, we also demonstrate that the chirality-dependent spin-polarized MZMs and LAR processes could be realized in the experimentally accessible 1D helical ${\mathrm{NbSe}}_{\text{3}}$ nanowire. Thus, our findings offer insights into the realization and manipulation of MZMs in 1D chiral systems, with implications for spintronic applications in TSC-based devices.

A Majorana Fermion (MF), defined as a particle that is its own anti-particle, can be engineered in the form of quasi-particles appearing in condensed matter systems. Majorana zero modes (MZMs), topologically protected zero-energy states, can exist at the topological phase boundary of one- or two-dimensional systems. These MZMs follow non-Abelian statistics and can be used as a building block for fault-tolerant quantum computers. Here, we introduce the conceptual development of MZMs based on Kitaev’s chain model and give a brief history of the experimental realization of MZMs.

Probing Majorana zero modes by measuring transport through an interacting magnetic impurity

Three-dimensional (3D) Dirac semimetals, which possess 3D linear dispersion in the electronic structure as a bulk analogue of graphene, have lately generated widespread interest in both materials science and condensed matter physics. Recently, crystalline Cd3As2 has been proposed and proved to be a 3D Dirac semimetal that can survive in the atmosphere. Here, by using point contact spectroscopy measurements, we observe exotic superconductivity around the point contact region on the surface of Cd3As2 crystals. The zero-bias conductance peak (ZBCP) and double conductance peaks (DCPs) symmetric around zero bias suggest p-wave-like unconventional superconductivity. Considering the topological properties of 3D Dirac semimetals, our findings may indicate that Cd3As2 crystals under certain conditions could be topological superconductors, which are predicted to support Majorana zero modes or gapless Majorana edge/surface modes in the boundary depending on the dimensionality of the material.

The relation between band topology and Majorana zero energy modes (MZMs) in topological superconductors had been well studied in the past decades. However, the relation between the quantum metric and MZMs has yet to be understood. In this Letter, we first construct a three band Lieb-like lattice model with an isolated flat band and tunable quantum metric. By introducing nearest neighbor equal spin pairing, we obtain the Lieb-Kitaev model which supports MZMs. When the Fermi energy is set within the flat band energy, the MZMs, which are supposed to be well localized at the ends of the 1D superconductor due to the flatness of the band, appear. On the contrary, we show both numerically and analytically that the localization length of the MZMs is controlled by a length scale defined by the quantum metric of the flat band, which we call the quantum metric length (QML). The QML can be several orders of magnitude longer than the conventional BCS superconducting coherence length. When the QML is comparable to the length of the superconductor, the two MZMs from the two ends of the superconductor can hybridize and induce ultra long range crossed Andreev reflections. This Letter unveils how the quantum metric can greatly influence the properties of MZMs through the QML, and the results can be generalized to other topological bound states.

Despite recent experimental progress towards observing large zero-bias conductance peaks (ZBCPs) as signatures of Majorana modes, confusion remains about whether Majorana modes have been observed. This is in part due to the theoretical prediction of fine-tuned trivial (i.e., non-topological) zero-bias peaks that occur because of uncontrolled quantum dots or disorder potentials. While many aspects of the topological phase can be somewhat fine-tuned because the topological phase space is often small, the quantized height of the ZBCP associated with a Majorana mode is known to be robust at sufficiently low temperatures even as the tunnel barrier is pinched off to vanishingly small normal-state conductance. The key shortcoming of the existing experimental works is an acute lack of stability of the putative Majorana mode features, indicating the probable absence of a topological phase, and the current paper suggests specific experimentally accessible measures for a careful quantitative analysis of the measured ZBCP stability. In this paper, we study how the counter-intuitive robustness of the ZBCP height to the tunnel barrier strength can be used to distinguish Majorana modes from non-topological ZBCPs. To this end, we introduce a dimensionless quality factor $F$ to quantify the robustness of the ZBCP height based on the range of normal-state (i.e. above-gap) conductance (which depends crucially on the tunnel barrier height) over which the ZBCP height remains within a pre-specified range of quantization. By computing this quality factor $F$ together with the topological characteristics for a wide range of models and parameters, we find that Majoranas are significantly more robust (i.e., have a higher value of $F$) compared with non-topological ZBCPs in the ideal low-temperature limit. Even at a temperature as high as the experimentally used $20$ mK, we find that we can set a threshold value of $F\sim 2.5$ (for $\epsilon=0.1$) so that ZBCPs associated with a quality factor $F>2.5$ are likely topological and $F\ll 2.5$ are topologically trivial. More precisely, the value of $F$ is operationally related to the degree of separation of the Majorana modes in the system, although $F$ uses only the experimentally measured tunnel conductance properties. Finally, we discuss how the quality factor $F$ measured in a transport setup can help estimate the quality of topological qubits made from Majorana modes. In particular, we show that if the induced gap can be enhanced somehow beyond the currently available $\sim 30$ $\mu$eV in InAs/Al samples, large (small) values of $F$ could easily distinguish between stable topological (unstable trivial) ZBCPs with the quantum dot induced quasi-Majorana bound states occasionally behaving similar to topological Majorana modes in short wires.

Non-Abelian braiding is a key property of Majorana zero modes (MZMs) that can be utilized for topological quantum computation. However, the presence of trivial Andreev bound states (ABSs) in topological superconductors can hinder the non-Abelian braiding of MZMs. We systematically investigate the braiding properties of ABSs induced by various inhomogeneous potentials in nanowires and quantify the main obstacles to non-Abelian braiding. We find that if a trivial ABS is present at zero energy with a tiny energy fluctuation, the non-Abelian braiding property can be sustained for a larger braiding time cost since the undesired dynamic phase is suppressed. Under certain conditions, the non-Abelian braiding of ABSs can even surpass that of MZMs in realistic systems, suggesting that nearly zero energy ABSs with a tiny energy fluctuation might also be suitable for quantum computation.

Recent experimental progress introduced devices that can combine topological superconductivity with Coulomb-blockade effects. Experiments with these devices have already provided additional evidence for Majorana zero modes in proximity-coupled semiconductor wires. They also stimulated numerous ideas for how to exploit interactions between Majorana zero modes generated by Coulomb charging effects in networks of Majorana wires. Coulomb effects promise to become a powerful tool in the quest for a topological quantum computer as well as for driving topological superconductors into topologically ordered insulating states. Here, we present a focused review of these recent developments, including discussions of recent experiments, designs of topological qubits, Majorana-based implementations of universal quantum computation, and topological quantum error correction. Motivated by the analogy between a qubit and a spin-1/2 degree of freedom, we also review how coupling between Cooper-pair boxes leads to emergent topologically ordered insulating phases.

This graduate-level textbook is the first pedagogical synthesis of the field of topological insulators and superconductors, one of the most exciting areas of research in condensed matter physics. Presenting the latest developments, while providing all the calculations necessary for a self-contained and complete description of the discipline, it is ideal for graduate students and researchers preparing to work in this area, and it will be an essential reference both within and outside the classroom. The book begins with simple concepts such as Berry phases, Dirac fermions, Hall conductance and its link to topology, and the Hofstadter problem of lattice electrons in a magnetic field. It moves on to explain topological phases of matter such as Chern insulators, two- and three-dimensional topological insulators, and Majorana p-wave wires. Additionally, the book covers zero modes on vortices in topological superconductors, time-reversal topological superconductors, and topological responses/field theory and topological indices. The book also analyzes recent topics in condensed matter theory and concludes by surveying active subfields of research such as insulators with point-group symmetries and the stability of topological semimetals. Problems at the end of each chapter offer opportunities to test knowledge and engage with frontier research issues. Topological Insulators and Topological Superconductors will provide graduate students and researchers with the physical understanding and mathematical tools needed to embark on research in this rapidly evolving field.

We propose a systematic magnetic-flux-free approach to detect, manipulate, and braid Majorana fermions in a semiconductor-nanowire-based topological Josephson junction by utilizing the Majorana spin degree of freedom. We find an intrinsic $\ensuremath{\pi}$-phase difference between spin-triplet pairings enforced by the Majorana zero modes (MZMs) at the two ends of a one-dimensional spinful topological superconductor. This $\ensuremath{\pi}$ phase is identified to be a spin-dependent superconducting phase, referred to as the spin phase, which we show to be tunable by controlling spin-orbit coupling strength via electric gates. This electric controllable spin phase not only affects the coupling energy between MZMs but also leads to a fractional Josephson effect in the absence of any applied magnetic flux, which enables the efficient topological qubit readout. We thus propose an all-electrically controlled superconductor-semiconductor hybrid circuit to manipulate MZMs and to detect their non-Abelian braiding statistics properties. Our work on spin properties of topological Josephson effects potentially opens up a new thrust for spintronic applications with Majorana-based semiconductor quantum circuits.

The issue on the effect of interactions in topological states concerns not only interacting topological phases but also novel symmetry-breaking phases and phase transitions. Here we study the interaction effect on Majorana zero modes (MZMs) bound to a square vortex lattice in two-dimensional topological superconductors. Under the neutrality condition, where single-body hybridization between MZMs is prohibited by an emergent symmetry, a minimal square-lattice model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo simulation. Guided by an insight from a further duality mapping, we demonstrate that the interaction induces a Majorana stripe state, a gapped state spontaneously breaking lattice translational and rotational symmetries, as opposed to the previously conjectured topological quantum criticality. Away from neutrality, a mean-field theory suggests a quantum critical point induced by hybridization.

Topologically nontrivial Fe-based superconductors attract extensive attentions due to their ability of hosting Majorana zero modes (MZMs) which could be used for topological quantum computation. Topological defects such as vortex lines are required to generate MZMs. Here, we observe the robust edge states along the surface steps of CaKFe4As4. Remarkably, the tunneling spectra show a sharp zero-bias peak (ZBP) with multiple integer-quantized states at the step edge under zero magnetic field. We propose that the increasing hole doping around step edges may drive the local superconductivity into a state with possible spontaneous time-reversal symmetry breaking. Consequently, the ZBP can be interpreted as an MZM in an effective vortex in the superconducting topological surface state by proximity to the center of a tri-junction with different superconducting order parameters. Our results provide new insights into the interplay between topology and unconventional superconductivity, and pave a new path to generate MZMs without magnetic field.