Majorana fermion (MF) excitations in solid state system have non-Abelian statistics which is essential for topological quantum computation. Previous proposals to realize MF, however, generally requires fine-tuning of parameters. Here we explore a platform which avoids the fine-tuning problem, namely a ferromagnetic chain deposited on the surface of a spin-orbit coupled $s$-wave superconductor. We show that it generically supports zero-energy topological MF excitations near the two ends of the chain with minimal fine-tuning. Depending on the strength of the ferromagnetic moment in the chain, the number of MFs at each end, $n$, can be either one or two, and should be revealed by a robust zero-bias peak (ZBP) of height $2ne^2/h$ in scanning tunneling microscopy (STM) measurements which would show strong (weak) signals at the ends (middle) of the chain. The role of an approximate chiral symmetry which gives an integer topological invariant to the system is discussed.
Disorder is known to suppress the gap of a topological superconducting state that would support non-Abelian Majorana zero modes. In this paper, we study using the self-consistent Born approximation the robustness of the Majorana modes to disorder within a suitably extended Eilenberger theory, in which the spatial dependence of the localized Majorana wave functions is included. We find that the Majorana mode becomes delocalized with increasing disorder strength as the topological superconducting gap is suppressed. However, surprisingly, the zero bias peak seems to survive even for disorder strength exceeding the critical value necessary for closing the superconducting gap within the Born approximation.
We show that a closed quantum system driven through a quantum critical point with two rates $omega_1$ (which controls its proximity to the quantum critical point) and $omega_2$ (which controls the dispersion of the low-energy quasiparticles at the critical point) exhibits novel scaling laws for defect density $n$ and residual energy $Q$. We demonstrate suppression of both $n$ and $Q$ with increasing $omega_2$ leading to an alternate route to achieving near-adiabaticity in a finite time for a quantum system during its passage through a critical point. We provide an exact solution for such dynamics with linear drive protocols applied to a class of integrable models, supplement this solution with scaling arguments applicable to generic many-body Hamiltonians, and discuss specific models and experimental systems where our theory may be tested.
Contrary to the widespread belief that Majorana zero-energy modes, existing as bound edge states in 2D topological insulator (TI)-superconductor (SC) hybrid structures, are unaffected by non-magnetic static disorder by virtue of Andersons theorem, we show that such a protection against disorder does not exist in realistic multi-channel TI/SC/ferromagnetic insulator (FI) sandwich structures of experimental relevance since the time-reversal symmetry is explicitly broken locally at the SC/FI interface where the end Majorana mode (MM) resides. We find that although the MM itself and the emph{bulk} topological superconducting phase inside the TI are indeed universally protected against disorder, disorder-induced subgap states are generically introduced at the TI edge due to the presence of the FI/SC interface as long as multiple edge channels are occupied. We discuss the implications of the finding for the detection and manipulation of the edge MM in realistic TI/SC/FI experimental systems of current interest.
Using Bogoliubov-de Gennes (BdG) equations we numerically calculate the disorder averaged density of states of disordered semiconductor nanowires driven into a putative topological p-wave superconducting phase by spin-orbit coupling, Zeeman spin splitting and s-wave superconducting proximity effect induced by a nearby superconductor. Comparing with the corresponding theoretical self-consistent Born approximation (SCBA) results treating disorder effects, we comment on the topological phase diagram of the system in the presence of increasing disorder. Although disorder strongly suppresses the zero-bias peak (ZBP) associated with the Majorana zero mode, we find some clear remnant of a ZBP even when the topological gap has essentially vanished in the SCBA theory because of disorder. We explicitly compare effects of disorder on the numerical density of states in the topological and trivial phases.
Recent observations of a zero bias conductance peak in tunneling transport measurements in superconductor--semiconductor nanowire devices provide evidence for the predicted zero--energy Majorana modes, but not the conclusive proof for their existence. We establish that direct observation of a splitting of the zero bias conductance peak can serve as the smoking gun evidence for the existence of the Majorana mode. We show that the splitting has an oscillatory dependence on the Zeeman field (chemical potential) at fixed chemical potential (Zeeman field). By contrast, when the density is constant rather than the chemical potential -- the likely situation in the current experimental set-ups -- the splitting oscillations are generically suppressed. Our theory predicts the conditions under which the splitting oscillations can serve as the smoking gun for the experimental confirmation of the elusive Majorana mode.
Topological Majorana fermion (MF) quasiparticles have been recently suggested to exist in semiconductor quantum wires with proximity induced superconductivity and a Zeeman field. Although the experimentally observed zero bias tunneling peak and a fractional ac-Josephson effect can be taken as necessary signatures of MFs, neither of them constitutes a sufficient smoking gun experiment. Since one pair of Majorana fermions share a single conventional fermionic degree of freedom, MFs are in a sense fractionalized excitations. Based on this fractionalization we propose a tunneling experiment that furnishes a nearly unique signature of end state MFs in semiconductor quantum wires. In particular, we show that a teleportation-like experiment is not enough to distinguish MFs from pairs of MFs, which are equivalent to conventional zero energy states, but our proposed tunneling experiment, in principle, can make this distinction.
Topological superconductors supporting Majorana Fermions with non-abelian statistics are presently a subject of intense theoretical and experimental effort. It has been proposed that the observation of a half-frequency or a fractional Josephson effect is a more reliable test for topological superconductivity than the search for end zero modes. Low-energy end modes can occur accidentally due to impurities. In fact, the fractional Josephson effect has been observed for the semiconductor nanowire system. Here we consider the ac Josephson effect in a conventional s-wave superconductor-normal metal-superconductor junction at a finite voltage. Using a Floquet-Keldysh treatment of the finite voltage junction, we show that the power dissipated from the junction, which measures the ac Josephson effect, can show a peak at half (or even incommensurate fractions) of the Josephson frequency. A similar conclusion is shown to hold for the Shapiro step measurement. The ac fractional Josephson peak can also be understood simply in terms of Landau-Zener processes associated with the Andreev bound state spectrum of the junction.
Motivated by a recent experimental report[1] claiming the likely observation of the Majorana mode in a semiconductor-superconductor hybrid structure[2,3,4,5], we study theoretically the dependence of the zero bias conductance peak associated with the zero-energy Majorana mode in the topological superconducting phase as a function of temperature, tunnel barrier potential, and a magnetic field tilted from the direction of the wire for realistic wires of finite lengths. We find that higher temperatures and tunnel barriers as well as a large magnetic field in the direction transverse to the wire length could very strongly suppress the zero-bias conductance peak as observed in Ref.[1]. We also show that a strong magnetic field along the wire could eventually lead to the splitting of the zero bias peak into a doublet with the doublet energy splitting oscillating as a function of increasing magnetic field. Our results based on the standard theory of topological superconductivity in a semiconductor hybrid structure in the presence of proximity-induced superconductivity, spin-orbit coupling, and Zeeman splitting show that the recently reported experimental data are generally consistent with the existing theory that led to the predictions for the existence of the Majorana modes in the semiconductor hybrid structures in spite of some apparent anomalies in the experimental observations at first sight. We also make several concrete new predictions for future observations regarding Majorana splitting in finite wires used in the experiments.
Spin-orbit coupled superconductors are potentially interesting candidates for realizing topological and potentially non-Abelian states with Majorana Fermions. We argue that time-reversal broken spin-orbit coupled superconductors generically can be characterized as having sub-gap states that are bound to localized non-magnetic impurities. Such bound states, which are referred to as Shiba states, can be detected as sharp resonances in the tunneling spectrum of the spin-orbit coupled superconductors. The Shiba state resonance can be tuned using a gate-voltage or a magnetic field from being at the edge of the gap at zero magnetic fields to crossing zero energy when the Zeeman splitting is tuned into the topological superconducting regime. The zero-crossing signifies a Fermion parity changing first order quantum phase transition, which is characterized by a Pfaffian topological invariant. These zero-crossings of the impurity level can be used to locally characterize the topological superconducting state from tunneling experiments.
