Chiral Majorana Edge Modes Research Articles

Anomalies-Rich Floquet superconductivities induced by joint modulation of dynamic driving and static parameters

Current theoretical and experimental endeavors to realize an anomalous Floquet chiral topological superconductor (TSC), which is characterized by chiral Majorana edge modes independent of the Chern number, remain insufficient. Herein, we propose a new scheme that involves jointly tuning dynamic driving and static parameters within a magnetic topological insulator-superconductor sandwich structure to achieve this goal. The Josephson phase modulation induced by an applied bias voltage across the structure is utilized as a Floquet periodic drive. It is found that the interplay between the two kinds of tunings can bring about a lot more exotic Floquet TSC phases than those caused by only tuning the dynamic driving parameter (frequency ω or period τ). More importantly, just tuning static parameters (the chemical potential µ, Zeeman field gz , and proximity-induced superconducting energy gap Δb ) also can induce a series of novel topological phase transitions. Particularly, the features in the context of the three tunings are different from each other, originating from the combination of intrinsic and different extrinsic mechanisms. In addition, jointly tuning τ and µ (gz ) can have its own unique TSC phases. The proposed scheme should be readily accessible in experiments, and thus the family of anomalous Floquet TSC phases may be considerably enriched.

Non-Abelian topological superconductivity in maximally twisted double-layer spin-triplet valley-singlet superconductors

Recent studies point to an exotic spin-triplet valley-singlet (STVS) superconducting phase in certain two-valley electron liquids, including rhombohedral trilayer graphene, Bernal bilayer graphene and ZrNCl, which nevertheless admits only trivial topology. Here, we predict that upon twisting two layers of STVS superconductors, a chiral fpm {{{{{{{rm{i}}}}}}}}{f}^{{prime} }-wave superconducting phase emerges near the ‘maximal’ twist angle of 30∘ where the system becomes an extrinsic quasi-crystal with 12-fold tiling. The resulting composite hosts an odd number of chiral Majorana edge modes and a single non-Abelian Majorana zero mode (MZM) in the vortex core. Through detailed symmetry analysis and microscopic modelling, we demonstrate that the non-Abelian topological superconductivity (TSC) forms robustly near the maximal twist when the isolated Fermi pockets coalesce into a single connected Fermi surface in the moiré Brillouin zone. Our results establish the large-twist-angle engineering, with distinct underlying moiré physics from magic-angle graphene, as a viable route toward non-Abelian TSC.

Coulomb blockade of chiral Majorana and complex fermions far from equilibrium

We study charge transport in a single-electron transistor implemented as an interferometer such that the Coulomb-blockaded middle island contains a circular chiral Majorana or Dirac edge mode. We concentrate on the regime of small conductance and provide an asymptotic solution in the limit of high transport voltage exceeding the charging energy. The solution is achieved using an instantonlike technique. The distinctions between Majorana and Dirac cases appear when the tunnel junctions are unequal. The main difference is in the offset current at high voltages which can be higher, up to 50% in the Majorana case. It is caused by an additional particle-hole symmetry of the distribution function of Majorana fermions. There is also an eye-catching distinction between the oscillation patterns of the current as a function of the gate charge. We conjecture this distinction survives at lower transport voltages as well.

Multilayer graphenes as a platform for interaction-driven physics and topological superconductivity

Motivated by the recent discoveries of superconductivity in bilayer and trilayer graphene, we theoretically investigate superconductivity and other interaction-driven phases in multilayer graphene stacks. To this end, we study the density of states of multilayer graphene with up to four layers at the single-particle band structure level in the presence of a transverse electric field. Among the considered structures, tetralayer graphene with rhombohedral (ABCA) stacking reaches the highest density of states. We study the phases that can arise in ABCA graphene by tuning the carrier density and transverse electric field. For a broad region of the tuning parameters, the presence of strong Coulomb repulsion leads to a spontaneous spin and valley symmetry breaking via Stoner transitions. Using a model that incorporates the spontaneous spin and valley polarization, we explore the Kohn-Luttinger mechanism for superconductivity driven by repulsive Coulomb interactions. We find that the strongest superconducting instability is in the $p-$wave channel, and occurs in proximity to the onset of Stoner transitions. Interestingly, we find a range of densities and transverse electric fields where superconductivity develops out of a strongly corrugated, singly-connected Fermi surface in each valley, leading to a topologically non-trivial chiral $p+ip$ superconducting state with an even number of co-propagating chiral Majorana edge modes. Our work establishes ABCA stacked tetralayer graphene as a promising platform for observing strongly correlated physics and topological superconductivity.

Topological phases and tunable Majorana zero modes in Rashba superconducting systems in the presence of Zeeman field

In the framework of the extended Bogoliubov-de Gennes theory, topological phase transitions and Majorana zero modes are investigated by diagonalizing the tight-binding model Hamiltonian for two-dimensional superconducting systems with Rashba spin-orbit coupling and spin correlations when the effect of Zeeman field is involved. With increasing the Zeeman-field strength, the first-order topological superconductivity and chiral Majorana edge modes may develop for the long Rashba superconducting stripe under appropriate chemical potential. Moreover, the second-order topological phase can be found for the Rashba loop-frame geometry. Relying on the chosen chemical potential near half filling, Majorana zero-energy states located at corners of the square loop are more feasible in the weak exchange-field regime. Particularly, the transitions between Majorana edge and corner modes can take place in the loop system when the field strength is varied. In addition, the number and location of robust Majorana corner modes are highly sensitive to the introduced surface defects. Our theoretical predictions may provide useful guidance for observing and tuning Majorana zero modes in future experiments and applications.

Directly probing the chirality of Majorana edge states

We propose to directly probe the chirality of Majorana edge states in 2D topological superconductors using polarization selective photon absorption. When shining circularly polarized light on a 2D topological superconductor in disk geometry, the photons can excite quasiparticles only when the polarization of the light matches the chirality of the Majorana edge states required by the angular momentum conservation. Hence, one can obtain the chirality of the Majorana edge states by measuring the photon absorption rate. We show that the polarization selective photon absorption can also serve as smoking gun evidence of the chiral Majorana edge mode.

Topological surface superconductivity in FeSe0.45Te0.55

The engineering of Majorana zero modes in topological superconductors, a paradigm for the realization of topological quantum computing and topology-based devices, has been hampered by the absence of materials with sufficiently large superconducting gaps. Recent experiments, however, have provided enthralling evidence for the existence of topological surface superconductivity in the iron-based superconductor FeSe0.45Te0.55 possessing a full s±-wave gap of a few meV. Here, we propose a mechanism for the emergence of topological superconductivity on the surface of FeSe0.45Te0.55 by demonstrating that the interplay between the s±-wave symmetry of the superconducting gap, surface magnetism, and a Rashba spin–orbit interaction gives rise to robust topological superconducting phases. Moreover, the proposed mechanism explains a series of experimentally observed hallmarks of topological superconductivity, such as the emergence of Majorana zero modes in the center of vortex cores and at the end of line defects, as well as of chiral Majorana edge modes along domain walls. We also propose that the spatial distribution of supercurrents near a domain wall is a characteristic signature measurable via a scanning superconducting quantum interference device that can distinguish between chiral Majorana edge modes and trivial in-gap states.

Shot noise and differential conductance as signatures of putative topological superconductivity in FeSe0.45Te0.55

We present a theory for the differential shot noise, $dS/dV$, as measured via shot-noise scanning tunneling spectroscopy, and the differential conductance, $dI/dV$, for tunneling into Majorana zero modes (MZMs) in the putative topological superconductor ${\mathrm{FeSe}}_{0.45}{\mathrm{Te}}_{0.55}$. We demonstrate that for tunneling into chiral Majorana edge modes near domain walls, as well as MZMs localized in vortex cores and at the end of defect lines, $dS/dV$ vanishes whenever $dI/dV$ reaches a quantized value being equal to the quantum of conductance. These results are independent of the particular orbital tunneling path, thus establishing a vanishing $dS/dV$ concomitant with a quantized $dI/dV$, as universal signatures for Majorana modes in two-dimensional topological superconductors, irrespective of the material's specific complex electronic band structure.

Proximity-induced superconducting gap in the intrinsic magnetic topological insulator MnBi2Te4

We report magnetotransport measurements in the NbN/ magnetic topological insulator MnBi2Te4 (MBT)/ NbN junction at low temperature. At 10 mK, the nonlinear current-voltage characteristic of the junction shows a tunneling behavior, indicating the existence of interfacial potential barriers within the heterostructure. Under an out of plane perpendicular magnetic field, a transition from negative to positive magnetoresistance (MR) is found when increasing the bias voltage. A proximity-induced superconducting gap is estimated to be 0.1meV by a pair of differential resistance dips. Moreover, the induced gap is enhanced by gradually tuning the Fermi level toward the charge neutral point by a back gate voltage, which is ascribed to the increased transport contribution of the topological surface states in MBT. Intriguingly, the induced gap exhibits an anomalous magnetic field assisted enhancement, which may originate from the spin orbit coupling and magnetic order of MBT. Our results reveal the interplay between magnetism and superconductivity in MBT, paving the way for further studies on topological superconductivity and chiral Majorana edge modes in quantum anomalous Hall insulator/superconductor hybrid systems.

Dissipationless spin current generation in a Kitaev chiral spin liquid

$\alpha$-RuCl$_3$ is a promising candidate material for the Kitaev spin liquid state where the half quantization of the thermal Hall effect, suggesting a topological character, has been observed. However, its relation to the existence of chiral Majorana fermions, which is predicted from the theory of the Kitaev model, is still under debate. Here we propose a more direct signature of a chiral Majorana edge mode which emerges in a universal scaling of the Drude weight of the edge spin Seebeck effect in the Kitaev model. Moreover, the absence of backscatterings in the chiral edge mode results in the generation of a dissipationless spin current in spite of an extremely short spin correlation length close to a lattice constant in the bulk. This result is not only experimentally observable, but also opens a way towards spintronics application of Kitaev materials. In contrast to magnetically ordered systems, a Kitaev spin liquid state is stable even in the atomic scale and makes it possible to fabricate a highly integrated device with substantial efficiency to generate a spin current. This proposal for using $\alpha$-RuCl$_3$ as a nano-scale device will pioneer Kitaev spintronics.

Stability of ordered and disordered phases in the Heisenberg-Kitaev model in a magnetic field

The S = 1/2 Kitaev honeycomb model has attracted significant attention as an exactly solvable example with a quantum spin liquid ground state. In an properly oriented external magnetic field, the system exhibits chiral Majorana edge modes with an associated quantized thermal Hall conductance, and a distinct spin-disordered phase emerges at intermediate field strengths, below the polarized phase. However, since material realizations of Kitaev magnetism invariably display competing exchange interactions, the stability of these exotic phases with respect to additional couplings is a key issue. Here, we report a 24-site exact diagonalization study of the Heisenberg-Kitaev model in a magnetic field applied in the [001] and [111] directions. By mapping the full phase diagram of the model and contrasting the results to recent nonlinear spin-wave calculations, we show that both methods agree well, thus establishing that quantum corrections substantially modify the classical phase diagram. Furthermore, we find that, in a [111] field, the intermediate-field spin-disordered phase is remarkably stable to Heisenberg interactions and may potentially end in a novel quantum tricritical point.

Thermal Probes of Phonon-Coupled Kitaev Spin Liquids: From Accurate Extraction of Quantized Edge Transport to Anyon Interferometry

Recent theoretical studies inspired by experiments on the Kitaev magnet $\alpha$-RuCl$_3$ highlight the nontrivial impact of phonons on the thermal Hall conductivity of chiral topological phases. Here we introduce mixed mesoscopic-macroscopic devices that allow refined thermal-transport probes of non-Abelian spin liquids with Ising topological order. These devices feature a quantum-coherent mesoscopic region with negligible phonon conductance, flanked by macroscopic lobes that facilitate efficient thermalization between chiral Majorana edge modes and bulk phonons. We show that our devices enable $(i)$ accurate determination of the quantized thermal Hall conductivity, $(ii)$ identification of non-Abelian Ising anyons via the temperature dependence of the thermal conductance, and most interestingly $(iii)$ single-anyon detection through heat-based anyon interferometry. Analogous results apply broadly to phonon-coupled chiral topological orders.

Chiral Majorana edge modes and vortex Majorana zero modes in superconducting antiferromagnetic topological insulator

The antiferromagnetic topological insulator (AFTI) is topologically protected by the combined time-reversal and translational symmetry . In this paper we investigate the effects of the s-wave superconducting pairings on the multilayers of AFTI, which breaks symmetry and can realize quantum anomalous Hall insulator with unit Chern number. For the weakly coupled pairings, the system corresponds to the topological superconductor (TSC) with the Chern number C = ±2. We answer the following questions whether the local Chern numbers and chiral Majorana edge modes of such a TSC distribute around the surface layers. By the numerical calculations based on a theoretic model of AFTI, we find that when the local Chern numbers are always dominated by the surface layers, the wavefunctions of chiral Majorana edge modes must not localize on the surface layers and show a smooth crossover from spatially occupying all layers to only distributing near the surface layers, similar to the hinge states in a three dimensional second-order topological phases. The latter phase, denoted by the hinged TSC, can be distinguished from the former phase by the measurements of the local density of state. In addition we also study the superconducting vortex phase transition in this system and find that the exchange field in the AFTI not only enlarges the phase space of topological vortex phase but also enhances its topological stability. These conclusions will stimulate the investigations on superconducting effects of AFTI and drive the studies on chiral Majorana edge modes and vortex Majorana zero modes into a new era.

Converting electrons into emergent fermions at a superconductor–Kitaev spin liquid interface

We study an interface between a Kitaev spin liquid (KSL) in the chiral phase and a non-chiral superconductor. When the coupling across the interface is sufficiently strong, the interface undergoes a transition into a phase characterized by a condensation of a bound state of a Bogoliubov quasiparticle in the superconductor and an emergent fermionic excitation in the spin liquid. In the condensed phase, electrons in the superconductor can coherently convert into emergent fermions in the spin liquid and vice versa. As a result, the chiral Majorana edge mode of the spin liquid becomes visible in the electronic local density of states at the interface, which can be measured in scanning tunneling spectroscopy experiments. We demonstrate the existence of this phase transition, and the non-local order parameter that characterizes it, using density matrix renormalization group simulations of a KSL strip coupled at its edge to a superconductor. An analogous phase transition can occur in a simpler system composed of a one-dimensional spin chain with a spin-flip $\mathbb{Z}_2$ symmetry coupled to a superconductor.

Prediction of topological superconductivity from type-IV, -III, -II, and -I′ nodal points induced by Rashba spin-orbit coupling

Topological superconductivity (TSC) has received great theoretical and experimental attention recently. Type-I Rashba nodal point (RNP) with isotropic band dispersions and point Fermi surface (FS) induced by the Rashba spin-orbit coupling (SOC) provides a promising route to the artificial TSC, because the inherent interspin coupling (ISC) shares identical form as the $p$-wave pairing $({k}_{x}{\ensuremath{\sigma}}_{y}\ensuremath{-}{k}_{y}{\ensuremath{\sigma}}_{x})$ exactly. Here we discuss the potential TSC of other types of RNPs with different ISC forms. By constructing a generic tight-binding model with Rashba SOC, we demonstrate type-IV, -III, -II, and -I\ensuremath{'} RNPs can be achieved on two-dimensional (2D) Bravais lattices, whose FS consists of only a hole (electron) pocket, two contacted hole (electron) pockets, contacted hole and electron pockets, and point of tangency, respectively. With the coorpration of $s$-wave pairing and Zeeman gaps, these new types of RNP will evoke TSC phases with chiral Majorana edge modes (MEMs), where the Chern number will be larger than 1 for multiple symmetry-equivalent RNPs. The Chern number can be further composited when the energies of unequivalent RNPs are equal, leading to edge-dependent MEMs. Moreover, by using first-principles calculations, we demonstrate the BiSb monolayer is an ideal platform for realizing TSC with Chern number 6 from type-II, -I, or -IV RNP. This work enriches the types of nodal point induced by Rashba SOC and offers a generic guidance on realizing multiple and edge-dependent MEMs from the abundantly synthesized 2D surface metal layers.

Possible topological phases in quantum anomalous Hall insulator/unconventional superconductor hybrid systems

Quantum anomalous Hall insulator (QAH)/$s$-wave superconductor (SC) hybrid systems are known to be an ideal platform for realizing two-dimensional topological superconductors with chiral Majorana edge modes. In this paper we study QAH/unconventional SC hybrid systems whose pairing symmetry is $p$-wave, $d$-wave, chiral $p$-wave, or chiral $d$-wave. The hybrid systems are a generalization of the QAH/$s$-wave SC hybrid system. In view of symmetries of the QAH and pairings, we introduce three topological numbers to classify topological phases of the hybrid systems. One is the Chern number that characterizes chiral Majorana edge modes and the others are topological numbers associated with crystalline symmetries. We numerically calculate the topological numbers and associated surface states for three characteristic regimes that feature an influence of unconventional SCs on QAHs. Our calculation shows a rich variety of topological phases and unveils the following topological phases that are no counterpart of the $s$-wave case: crystalline symmetry-protected helical Majorana edge modes, a line node phase (crystalline-symmetry-protected Bogoliubov Fermi surface), and multiple chiral Majorana edge modes. The phenomena result from a nontrivial topological interplay between the QAH and unconventional SCs. Finally, we discuss tunnel conductance in a junction between a normal metal and the hybrid systems, and show that the chiral and helical Majorana edge modes are distinguishable in terms of the presence/absence of zero-bias conductance peak.

Disorder-induced chiral and helical Majorana edge modes in a two-dimensional Ammann-Beenker quasicrystal

Recent research on disorder effects in topological phases in quasicrystalline systems has received much attention. In this work, by numerically computing the (spin) Bott index and the thermal conductance, we reveal the effects of disorder on a class D chiral topological superconductor and a class DIII time-reversal-invariant topological superconductor in a two-dimensional Ammann-Beenker tiling quasicrystalline lattice. We demonstrate that both the topologically protected chiral and helical Majorana edge modes are robust against weak disorder in the quasicrystalline lattice. More fascinating is the discovery of disorder-induced topologically nontrivial phases exhibiting chiral and helical Majorana edge modes in class D and DIII topological superconductor systems, respectively. Our findings open the door for the research on disorder-induced Majorana edge modes in quasicrystalline systems.

Anomalous Floquet Chiral Topological Superconductivity in a Topological Insulator Sandwich Structure.

We show that Floquet chiral topological superconductivity arises naturally in Josephson junctions made of magnetic topological insulator-superconductor sandwich structures. The Josephson phase modulation associated with an applied bias voltage across the junction drives the system into the anomalous Floquet chiral topological superconductor hosting chiral Majorana edge modes in the quasienergy spectrum, with the bulk Floquet bands carrying zero Chern numbers. The bias voltage acts as a tuning parameter enabling novel Floquet topological quantum phase transitions driving the system into a myriad of exotic Majorana-carrying Floquet topological superconducting phases. Our theory establishes a new paradigm for realizing Floquet chiral topological superconductivity in solid-state systems, which should be experimentally directly accessible.

Chiral topological superconducting state with Chern number C = −2 in Pb3Bi/Ge(111)

Materials realization of chiral topological superconductivity is a crucial condition for observing and manipulating Majorana fermions in condensed matter physics. Here we develop a tight-binding description of Pb$_3$Bi/Ge(111), identified recently as an appealing candidate system for realizing chiral $p$-wave topological superconductivity [Nat. Phys. 15, 796 (2019)]. We first show that our phenomenological model can capture the two main features of the electronic band structures obtained from first-principles calculations, namely, the giant Rashba splitting and type-II van Hove singularity. Next, when the $s$-wave superconducting property of the parent Pb system is explicitly considered, we find the alloyed system can be tuned into a chiral topological superconductor with Chern number $\mathcal{C} = -2$, resulting from the synergistic effect of a sufficiently strong Zeeman field and the inherently large Rashba spin-orbit coupling. The nontrivial topology with $\mathcal{C} = -2$ is further shown to be detectable as two chiral Majorana edge modes propagating along the same direction of the system with proper boundaries. We finally discuss the physically realistic conditions to establish the predicted topological superconductivity and observe the corresponding Majorana edge modes, including the influence of the superconducting gap, Land\'{e} $g$-factor, and critical magnetic field. The present study provides useful guides in searching for effective $p$-wave superconductivity and Majorana fermions in two-dimensional or related interfacial systems.

Local Raman spectroscopy of chiral Majorana edge modes in Kitaev spin liquids and topological superconductors

The Raman scattering with local optical excitation from the Majorana edge modes of Kitaev spin liquids and topological superconductors is studied theoretically. Although the effective one-dimensional model is common between these two cases, the coupling to the electromagnetic field is different. It is found that the Raman spectrum at low energy scales with $\omega^3$ in Kitaev spin liquids while it shows the gap in topological superconductors. This is in sharp contrast to the infrared absorption, where the spectrum shows the gap in Kitaev spin liquids, while it behaves as $\sim \omega^2$ in topological superconductors. This indicates that the electrodynamics of Majorana edge modes depends on their higher-dimensional origins. The realistic estimate of the Raman scattering intensity is given for $\alpha$-RuCl$_3$ as the candidate for Kitaev spin liquid.