Majorana Corner Modes Research Articles

Erratum: Braiding Majorana corner modes in a second-order topological superconductor [Phys. Rev. Research 2 , 032068(R) (2020)

Erratum: Braiding Majorana corner modes in a second-order topological superconductor [Phys. Rev. Research <b>2</b> , 032068(R) (2020)

Edge-dependent Majorana corner modes in an s-wave superconductor

In contrast to Majorana edge modes in first-order topological superconductors, Majorana corner modes are usually related to spatial symmetries and further rely on specific corner of the sample. In this paper, we study Majorana corner modes in an extended Kane-Mele model with arbitrary phase. Majorana corner mode exists at both corner between zigzag edge and armchair edge and corner between zigzag edge and zigzag edge for the phase at π/2. However, while the Majorana corner mode at the corner between zigzag edge and armchair edge remains as the phase deviates from π/2, Majorana mode at the corner between zigzag edge and armchair edge disappears. These Majorana corner modes are edge dependent and originate from the sign change of Dirac mass or Dirac velocity for edge states at adjacent edges tuned by the phase.

Machine learning guided discovery of stable, spin-resolved topological insulators

Identification of a nontrivial Z2 index in a spinful two-dimensional insulator indicates the presence of an odd, quantized (pseudo)spin-resolved Chern number, Cs=(C↑−C↓)/2. However, the statement is not biconditional. An odd spin-Chern number can survive when the familiar Z2 index vanishes. Identification of solid-state systems hosting an odd, quantized Cs and trivial Z2 index is a pressing issue due to the potential for such insulators to admit band gaps optimal for experiments and quantum devices. Nevertheless, they have proven elusive due to the computational expense associated with their discovery. In this work, a neural network capable of identifying the spin-Chern number is developed and used to identify the first solid-state systems hosting a trivial Z2 index and odd Cs. We demonstrate the potential of one such system, Ti2CO2, to support Majorana corner modes via the superconducting proximity effect. Published by the American Physical Society 2024

Interaction-driven first-order and higher-order topological superconductivity

We investigate topological superconductivity in the Rashba-Hubbard model, describing heavy-atom superlattice and van der Waals materials with broken inversion. We focus in particular on fillings close to the van Hove singularities, where a large density of states enhances the superconducting transition temperature. To determine the topology of the superconducting gaps and to analyze the stability of their surface states in the presence of disorder and residual interactions, we employ an fRG+MFT approach, which combines the unbiased functional renormalization group (fRG) with a real-space mean-field theory (MFT). Our approach uncovers a cascade of topological superconducting states, including A1 and B1 pairings, whose wave functions are of dominant p- and d-wave character, respectively, as well as a time-reversal breaking A1+iB1 pairing. While the A1 and B1 states have first-order topology with helical and flat-band Majorana edge states, respectively, the A1+iB1 pairing exhibits second-order topology with Majorana corner modes. We investigate the disorder stability of the bulk superconducting states, analyze interaction-induced instabilities of the edge states, and discuss implications for experimental systems. Published by the American Physical Society 2024

Majorana corner modes and tunable patterns in an altermagnet heterostructure

Majorana corner modes and tunable patterns in an altermagnet heterostructure

Majorana Corner Modes and Flat-Band Majorana Edge Modes in Superconductor/Topological-Insulator/Superconductor Junctions

Recently, superconductors with higher-order topology have stimulated extensive attention and research interest. Higher-order topological superconductors exhibit unconventional bulk-boundary correspondence, thus allow exotic lower-dimensional boundary modes, such as Majorana corner and hinge modes. However, higher-order topological superconductivity has yet to be found in naturally occurring materials. We investigate higher-order topology in a two-dimensional Josephson junction comprised of two s-wave superconductors separated by a topological insulator thin film. We find that zero-energy Majorana corner modes, a boundary fingerprint of higher-order topological superconductivity, can be achieved by applying magnetic field. When an in-plane Zeeman field is applied to the system, two corner modes appear in the superconducting junction. Furthermore, we also discover a two-dimensional nodal superconducting phase which supports flat-band Majorana edge modes connecting the bulk nodes. Importantly, we demonstrate that zero-energy Majorana corner modes are stable when increasing the thickness of topological insulator thin film.

Perfectly localized Majorana corner modes in fermionic lattices

Perfectly localized Majorana corner modes in fermionic lattices

High-temperature Majorana corner modes in a d+id′ superconductor heterostructure: Application to twisted bilayer cuprate superconductors

The realization of Majorana corner modes generally requires unconventional superconducting pairing or $s$-wave pairing. However, the bulk nodes in unconventional superconductors and the low $T_c$ of $s$-wave superconductors are not conducive to the experimental observation of Majorana corner modes. Here we show the emergence of a Majorana corner mode at each corner of a two-dimensional topological insulator in proximity to a $d+id'$ pairing superconductor, such as heavily doped graphene or especially a twisted bilayer of a cuprate superconductor, e.g., Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, which has recently been proposed as a fully gapped chiral $d_{x^2-y^2}+id_{xy}$ superconductor with $T_c$ close to its native 90 K, and an in-plane magnetic field. By numerical calculation and intuitive edge theory, we find that the interplay of the proximity-induced pairing and Zeeman field can introduce opposite Dirac masses on adjacent edges of the topological insulator, which creates one zero-energy Majorana mode at each corner. Our scheme offers a feasible route to achieve and explore Majorana corner modes in a high-temperature platform without bulk superconductor nodes.

Higher order topological superconductivity in magnet-superconductor hybrid systems

Quantum engineering of topological superconductors and of the ensuing Majorana zero modes might hold the key for realizing topological quantum computing and topology-based devices. Magnet-superconductor hybrid (MSH) systems have proven to be experimentally versatile platforms for the creation of topological superconductivity by custom-designing the complex structure of their magnetic layer. Here, we demonstrate that higher order topological superconductivity (HOTSC) can be realized in two-dimensional MSH systems by using stacked magnetic structures. We show that the sensitivity of the HOTSC to the particular magnetic stacking opens an intriguing ability to tune the system between trivial and topological phases using atomic manipulation techniques. We propose that the realization of HOTSC in MSH systems, and in particular the existence of the characteristic Majorana corner modes, allows for the implementation of a measurement-based protocols for topological quantum computing.

Tunable boson-assisted finite-range interaction and engineering Majorana corner modes in optical lattices

Nonlocal interaction between ultracold atoms trapped in optical lattices can give rise to interesting quantum many-body phenomena. However, its realization usually demands unconventional techniques, for example, the artificial gauge fields or higher-orbit Feshbach resonances, and is not highly controllable. Here, we propose a valid and feasible scheme for realizing a tunable finite-range interaction for spinless fermions immersed into the bath of bosons. The strength of the effective interaction for the fermionic subsystem is artificially tunable by manipulating bosons, ranging from the repulsive to the attractive regime. In addition, the interaction distance is locked to the hopping of bosons, making the finite-range interaction perfectly clean for the fermionic subsystem. Specifically, we find that, by introducing an additional staggered hopping of bosons, the proposal is readily applied to search the Majorana corner modes in such a spinless system, without the implementation of complex artificial gauge fields, which is totally distinct from existing results reported in spinful systems. Therefore, this scheme provides a potential platform for exploring the unconventional topological superfluids and other nontrivial phases induced by long-range interactions in ultracold atoms.

Effect of local Coulomb interaction on Majorana corner modes: Weak and strong correlation limits

Here we present an analysis of the evolution of Majorana corner modes realizing in a higher-order topological superconductor (HOTSC) on a square lattice under the influence of local Coulomb repulsion. The HOTSC spectral properties were considered in two regimes: when the intensities of many-body interactions are either weak or strong. The weak regime was studied using the mean-field approximation with self-consistent solutions carried out both in the uniform case and taking into account of the boundary of the finite square-shaped system. It is shown that in the uniform case the topologically nontrivial phase on the phase diagram is widened by the Coulomb repulsion. The boundary effect, resulting in an inhomogeneous spatial distribution of the correlators, leads to the appearance of the crossover from the symmetric spin-independent solution to the spin-dependent one characterized by a spontaneously broken symmetry. In the former the corner states have energies that are determined by the overlap of the excitation wave functions localized at the different corners. In the latter the corner excitation energy is defined by the Coulomb repulsion intensity with a quadratic law. The crossover is a finite size effect, i.e. the larger the system the lesser the critical value of the Coulomb repulsion. In the strong repulsion regime we derive the effective HOTSC Hamiltonian in the atomic representation and found a rich variety of interactions induced by virtual processes between the lower and upper Hubbard subbands. It is shown that Majorana corner modes still can be realized in the limit of the infinite repulsion. Although the boundaries of the topologically nontrivial phase are strongly renormalized by Hubbard corrections.

Time evolution of Majorana corner modes in a Floquet second-order topological superconductor

We propose a practically feasible time-periodic sinusoidal drive protocol in onsite mass term to generate the two-dimensional~(2D) Floquet second-order topological superconductor, hosting both the regular $0$- and anomalous $\pi$-Majorana corner modes~(MCMs) while starting from a static 2D topological insulator/$d$-wave superconductor heterostructure setup. We theoretically study the local density spectra and the time dynamics of MCMs in the presence of such drive. The dynamical MCMs are topologically characterized by employing the average quadrupolar motion. Furthermore, we employ the Floquet perturbation theory~(FPT) in the strong driving amplitude limit to provide analytical insight into the problem. We compare our exact (numerical), and the FPT results in terms of the eigenvalue spectra and the time dynamics of the MCMs. We emphasize that the agreement between the exact numerical and the FPT results are more prominent in the higher frequency regime for close to the $0$-quasi-energy mode.

О дираковской массе хаббардовских фермионов в сильно коррелированном топологическом сверхпроводнике высокого порядка

Majorana corner modes have a number of advantages over the classical Majorana states in terms of performing topologically protected quantum computations. However, the problem of the influence of Coulomb repulsion on higher-order phases, which inevitably arises when trying to implement such systems in practice, has been poorly studied. In this article, we analyze the features of a topological invariant describing the nontrivial phase with the corner modes for a two-dimensional two-orbital model of a hybrid structure in the regime of extremely strong electron correlations. For this purpose, approximate wave functions of the edge states with a linear dispersion law and the associated Dirac masses, which arise when superconducting pairing in the system is taken into account, are obtained.

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.

Engineering Majorana corner modes from two-dimensional hexagonal crystals

Second order topological insulator can be engineered from two-dimensional materials with strong spin-orbit coupling and in-plane Zeeman field. In proximity to superconductor, topological superconducting phase could be induced in the two-dimensional materials, which host Majorana corner modes at the intersection between two zigzag edges. Two types of tight binding models in hexagonal lattice, which include $p_{z}$ or $p_{x,y}$ orbit(s) in each lattice site, are applied to engineer two-dimensional materials in topological superconducting phase. In both models, the condition that induces the second order topological superconductor requires nonuniform value of either in-plane Zeeman fields or superconductor pairing parameters in two sublattices. The finite size effect of the second model is weaker than that of the first model.

Generating Many Majorana Corner Modes and Multiple Phase Transitions in Floquet Second-Order Topological Superconductors

A d-dimensional, nth-order topological insulator or superconductor has localized eigenmodes at its (d−n)-dimensional boundaries (n≤d). In this work, we apply periodic driving fields to two-dimensional superconductors, and obtain a wide variety of Floquet second-order topological superconducting (SOTSC) phases with many Majorana corner modes at both zero and π quasienergies. Two distinct Floquet SOTSC phases are found to be separated by three possible kinds of transformations, i.e., a topological phase transition due to the closing/reopening of a bulk spectral gap, a topological phase transition due to the closing/reopening of an edge spectral gap, or an entirely different phase in which the bulk spectrum is gapless. Thanks to the strong interplay between driving and intrinsic energy scales of the system, all the found phases and transitions are highly controllable via tuning a single hopping parameter of the system. Our discovery not only enriches the possible forms of Floquet SOTSC phases, but also offers an efficient scheme to generate many coexisting Majorana zero and π corner modes, which may find applications in Floquet quantum computation.

Geometric effect on Majorana corner modes in mesoscopic superconducting loop systems with Rashba spin-orbit interaction

Geometric effect on Majorana corner modes in mesoscopic superconducting loop systems with Rashba spin-orbit interaction

Majorana corner modes in mesoscopic superconducting square systems with mixed pairing in the presence of spin–orbit interaction

Majorana corner modes in mesoscopic superconducting square systems with mixed pairing in the presence of spin–orbit interaction

Effective Model for Fractional Topological Corner Modes in Quasicrystals.

High-order topological insulators (HOTIs), as generalized from topological crystalline insulators, are characterized with lower-dimensional metallic boundary states protected by spatial symmetries of a crystal, whose theoretical framework based on band inversion at special k points cannot be readily extended to quasicrystals because quasicrystals contain rotational symmetries that are not compatible with crystals, and momentum is no longer a good quantum number. Here, we develop a low-energy effective model underlying HOTI states in 2D quasicrystals for all possible rotational symmetries. By implementing a novel Fourier transform developed recently for quasicrystals and approximating the long-wavelength behavior by their large-scale average, we construct an effective k·p Hamiltonian to capture the band inversion at the center of a pseudo-Brillouin zone. We show that an in-plane Zeeman field can induce mass kinks at the intersection of adjacent edges of a 2D quasicrystal topological insulators and generate corner modes (CMs) with fractional charge, protected by rotational symmetries. Our model predictions are confirmed by numerical tight-binding calculations. Furthermore, when the quasicrystal is proximitized by an s-wave superconductor, Majorana CMs can also be created by tuning the field strength and chemical potential. Our work affords a generic approach to studying the low-energy physics of quasicrystals, in association with topological excitations and fractional statistics.

Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study

Coupled-wire constructions have been widely applied to quantum Hall systems and symmetry-protected topological (SPT) phases. In this Letter, we use the coupled one-dimensional nonchiral Luttinger liquids with domain-wall structured mass terms as quantum wires to construct the crystalline higher-order topological superconductors (HOTSC) in two-dimensional interacting fermionic systems by two representative examples: $D_4$-symmetric class-D HOTSC and $C_4$-symmetric class-BD\1 HOTSC, with Majorana corner modes on the edge. Furthermore, based on the coupled-wire constructions, the quantum phase transition between different phases of 2D HOTSC by tuning the inter-wire couplings is investigated in a straightforward way.