Majorana zero modes induced by superconducting phase bias

Majorana zero modes in condensed matter systems have been the subject of much interest in recent years. Their non-Abelian exchange statistics, making them a unique state of matter, and their potential applications in topological quantum computation, earned them attention from both theorists and experimentalists. It is generally understood that in order to form Majorana zero modes in quasi-one-dimensional topological insulators, time-reversal symmetry must be broken. The straightforward mechanisms for doing so—applying magnetic fields or coupling to ferromagnets—turned out to have many unwanted side effects, such as degradation of superconductivity and the formation of sub-gap states, which is part of the reason Majorana zero modes have been eluding direct experimental detection for a long time. Here we review several proposal that rely on controlling the phase of the superconducting order parameter, either as the sole mechanism for time-reversal-symmetry breaking, or as an additional handy knob used to reduce the applied magnetic field. These proposals hold practical promise to improve Majorana detection, and they shed light on the physics underlying the formation of the topological superconducting state.

In-plane magnetic field-driven symmetry breaking in topological insulator-based three-terminal junctions

Topological surface states of three-dimensional topological insulator nanoribbons and their distinct magnetoconductance properties are promising for topoelectronic applications and topological quantum computation. A crucial building block for nanoribbon-based circuits are three-terminal junctions. While the transport of topological surface states on a planar boundary is not directly affected by an in-plane magnetic field, the orbital effect cannot be neglected when the surface states are confined to the boundary of a nanoribbon geometry. Here, we report on the magnetotransport properties of such three-terminal junctions. We observe a dependence of the current on the in-plane magnetic field, with a distinct steering pattern of the surface state current towards a preferred output terminal for different magnetic field orientations. We demonstrate that this steering effect originates from the orbital effect, trapping the phase-coherent surface states in the different legs of the junction on opposite sides of the nanoribbon and breaking the left-right symmetry of the transmission across the junction. The reported magnetotransport properties demonstrate that an in-plane magnetic field is not only relevant but also very useful for the characterization and manipulation of transport in three-dimensional topological insulator nanoribbon-based junctions and circuits, acting as a topoelectric current switch.

A Majorana perspective on understanding and identifying axion insulators

An axion insulator is theoretically introduced to harbor unique surface states with half-integer Chern number $${{{{{{{\mathcal{C}}}}}}}}$$ C . Recently, experimental progress has been made in different candidate systems, while a unique Hall response to directly reflect the half-integer Chern number is still lacking to distinguish an axion state from other possible insulators. Here we show that the $${{{{{{{\mathcal{C}}}}}}}}=\frac{1}{2}$$ C = 1 2 axion state corresponds to a topological state with Chern number $${{{{{{{\mathcal{N}}}}}}}}=1$$ N = 1 in the Majorana basis. In proximity to an s − wave superconductor, a topological phase transition to an $${{{{{{{\mathcal{N}}}}}}}}=0$$ N = 0 phase takes place at critical superconducting pairing strength. Our theoretical analysis shows that a chiral Majorana hinge mode emerges at the boundary of $${{{{{{{\mathcal{N}}}}}}}}=1$$ N = 1 and $${{{{{{{\mathcal{N}}}}}}}}=0$$ N = 0 regions on the surface of an axion insulator. Furthermore, we propose a half-integer quantized thermal Hall conductance via a thermal transport measurement, which is a signature of the gapless chiral Majorana mode and thus confirms the $${{{{{{{\mathcal{C}}}}}}}}=\frac{1}{2}$$ C = 1 2 ($${{{{{{{\mathcal{N}}}}}}}}=1$$ N = 1 ) topological nature of an axion state. Our proposals help to theoretically comprehend and experimentally identify the axion insulator and may benefit the research of topological quantum computation.

Nano-photoluminescence of natural anyon molecules and topological quantum computation

The proposal of fault-tolerant quantum computations, which promise to dramatically improve the operation of quantum computers and to accelerate the development of the compact hardware for them, is based on topological quantum field theories, which rely on the existence in Nature of physical systems described by a Lagrangian containing a non-Abelian (NA) topological term. These are solid-state systems having two-dimensional electrons, which are coupled to magnetic-flux-quanta vortexes, forming complex particles, known as anyons. Topological quantum computing (TQC) operations thus represent a physical realization of the mathematical operations involving NA representations of a braid group Bn, generated by a set of n localized anyons, which can be braided and fused using a “tweezer” and controlled by a detector. For most of the potential TQC material systems known so far, which are 2D-electron–gas semiconductor structure at high magnetic field and a variety of hybrid superconductor/topological-material heterostructures, the realization of anyon localization versus tweezing and detecting meets serious obstacles, chief among which are the necessity of using current control, i.e., mobile particles, of the TQC operations and high density electron puddles (containing thousands of electrons) to generate a single vortex. Here we demonstrate a novel system, in which these obstacles can be overcome, and in which vortexes are generated by a single electron. This is a ~ 150 nm size many electron InP/GaInP2 self-organized quantum dot, in which molecules, consisting of a few localized anyons, are naturally formed and exist at zero external magnetic field. We used high-spatial-resolution scanning magneto-photoluminescence spectroscopy measurements of a set of the dots having five and six electrons, together with many-body quantum mechanical calculations to demonstrate spontaneous formation of the anyon magneto-electron particles (eν) having fractional charge ν = n/k, where n = 1–4 and k = 3–15 are the number of electrons and vortexes, respectively, arranged in molecular structures having a built-in (internal) magnetic field of 6–12 T. Using direct imaging of the molecular configurations we observed fusion and braiding of eν-anyons under photo-excitation and revealed the possibility of using charge sensing for their control. Our investigations show that InP/GaInP2 anyon-molecule QDs, which have intrinsic transformations of localized eν-anyons compatible with TQC operations and capable of being probed by charge sensing, are very promising for the realization of TQC.

Robustness of Majorana Zero-energy State in a Spin-orbit Coupled Semiconductor/superconductor Heterostructure

Based on the principle of linearized stability proposed by Lyapounov, we investigate the robustness of Majorana zero energy state (MZES), which plays an important role in topological quantum computation. Our study is different from previous works that usually explore the stability of MZES by the numerical test of some special perturbations, our treatment is suitable for arbitrary perturbations. Since our method follows the stability theory of differential equation, the results we obtained are reliable. As an example, we demonstrate it by the stability of MZES in the spin-orbit coupled semiconductor/ superconductor junction, the analytical and numerical results indicate that the MZES is unstable in this system.

Topological braiding and virtual particles on the cell membrane

Braiding of topological structures in complex matter fields provides a robust framework for encoding and processing information, and it has been extensively studied in the context of topological quantum computation. In living systems, topological defects are crucial for the localization and organization of biochemical signaling waves, but their braiding dynamics remain unexplored. Here, we show that the spiral wave cores, which organize the Rho-GTP protein signaling dynamics and force generation on the membrane of starfish egg cells, undergo spontaneous braiding dynamics. Experimentally measured world line braiding exponents and topological entropy correlate with cellular activity and agree with predictions from a generic field theory. Our analysis further reveals the creation and annihilation of virtual quasi-particle excitations during defect scattering events, suggesting phenomenological parallels between quantum and living matter.

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.

Assessing Bound States in a One-Dimensional Topological Superconductor: Majorana versus Tamm

Majorana bound states in topological superconductors have attracted intense research activity in view of applications in topological quantum computation. However, they are not the only example of topological bound states that can occur in such systems. Here, we study a model in which both Majorana and Tamm bound states compete. We show both numerically and analytically that, surprisingly, the Tamm state remains partially localized even when the spectrum becomes gapless. Despite this fact, we demonstrate that the Majorana polarization shows a clear transition between the two regimes.

Visibility of noisy quantum dot-based measurements of Majorana qubits

Measurement schemes of Majorana zero modes (MZMs) based on quantum dots (QDs) are of current interest as they provide a scalable platform for topological quantum computation. In a coupled qubit-QD setup we calculate the dependence of the charge of the QD and its differential capacitance on experimentally tunable parameters for both 2-MZM and 4-MZM measurements. We quantify the effect of noise on the measurement visibility by considering 1/f noise in detuning, tunneling amplitudes or phase. We find that on- or close-to-resonance measurements are generally preferable and predict, using conservative noise estimates, that noise coupling to the QDs is not a limitation to high-fidelity measurements of topological qubits.