Second-order Topological Superconductors Research Articles

Floquet generation of a second-order topological superconductor

We theoretically investigate the Floquet generation of a second-order topological superconducting (SOTSC) phase hosting Majorana corner modes (MCMs), considering a quantum spin Hall insulator with a proximity-induced superconducting $s$-wave pairing in it. Our dynamical prescription consists of the periodic kick in time-reversal symmetry breaking the in-plane magnetic field and fourfold rotational symmetry breaking the mass term in the bulk while these Floquet MCMs are preserved by antiunitary particle-hole symmetry. The first driving protocol always leads to four zero-energy MCMs (i.e., one Majorana state per corner) as a sign of a strong SOTSC phase. Interestingly, the second protocol can result in a weak SOTSC phase, harboring eight zero-energy MCMs (two Majorana states per corner), in addition to the strong SOTSC phase. We characterize the topological nature of these phases by a Floquet quadrupolar moment and Floquet Wannier spectrum. We believe that relying on the recent experimental advancement in the driven systems and proximity - superconductivity, our schemes may be possible to test in the future.

Quadrupole spin polarization as signature of second-order topological superconductors

We study theoretically second-order topological superconductors characterized by the presence of pairs of zero-energy Majorana corner states. We uncover a quadrupole spin polarization at the system edges that provides a striking signature to identify topological phases, thereby complementing standard approaches based on zero-bias conductance peaks due to Majorana corner states. We consider two different classes of second-order topological superconductors with broken time-reversal symmetry and show that both classes are characterized by a quadrupolar structure of the spin polarization that disappears as the system passes through the topological phase transition. This feature can be accessed experimentally using spin-polarized scanning tunneling microscopes. We study different models hosting second-order topological phases, both analytically and numerically, and using Keldysh techniques we provide numerical simulations of the spin-polarized currents probed by scanning tips.

Topological and holonomic quantum computation based on second-order topological superconductors

Majorana fermions feature non-Abelian exchange statistics and promise fascinating applications in topological quantum computation. Recently, second-order topological superconductors (SOTSs) have been proposed to host Majorana fermions as localized quasiparticles with zero excitation energy, pointing out a new avenue to facilitate topological quantum computation. We provide a minimal model for SOTSs and systematically analyze the features of Majorana zero modes with analytical and numerical methods. We further construct the fundamental fusion principles of zero modes stemming from a single or multiple SOTS islands. Finally, we propose concrete schemes in different setups formed by SOTSs, enabling us to exchange and fuse the zero modes for non-Abelian braiding and holonomic quantum gate operations.

Time-induced second-order topological superconductors

Higher-order topological materials with topologically protected states at the boundaries of their boundaries (hinges or corners) have attracted attention in recent years. In this paper, we utilize time-periodic driving to generate second-order topological superconductors out of systems which otherwise do not even allow second-order topological characterization. This is made possible by the design of the periodic drives which inherently exhibit nontrival winding in the time-domain. Through the interplay of topology in both spatial and temporal dimensions, nonchiral Majorana modes may emerge at the systems corners and sometimes even coexist with chiral Majorana modes. Our proposal thus presents a unique opportunity to Floquet engineering with minimal system complexity and its application in quantum information processing.

Braiding Majorana corner modes in a second-order topological superconductor

We propose the concept of a device based on a square-shaped sample of a two-dimensional second-order topological helical superconductor which hosts two zero-dimensional Majorana quasiparticles at the corners. The two zero-energy modes rely on particle-hole symmetry (PHS) and their spacial position can be shifted by rotating an in-plane magnetic field and tuning proximity-induced spin-singlet pairing. We consider an adiabatic cycle performed on the degenerate ground-state manifold and show that it realizes the braiding of the two modes whereby they accumulate a non-trivial statistical phase $\pi$ within one cycle. Alongside with the PHS-ensured operator algebra, the fractional statistics confirms the Majorana nature of the zero-energy excitations. A schematic design for a possible experimental implementation of such a device is presented, which could be a step towards realizing non-Abelian braiding.

All-electrically tunable networks of Majorana bound states

Second-order topological superconductors (SOTSs) host localized Majorana fermions and provide a new platform for topological quantum computation. We propose a remarkable and feasible way to realize networks based on SOTSs which allow to nucleate and braid Majorana bound states (MBSs) in an all-electrical manner without fine-tuning. The proposed setups are scalable in a straightforward way and can accommodate any even number of MBSs. Moreover, the MBSs in the networks allow defining qubits whose states can be initialized and read out by measuring Josephson currents flowing between SOTS islands. Our proposal can be implemented in monolayers of $\text{FeTe}{}_{1-x}\text{Se}_{x}$, monolayers of 1T'-WTe$_2$, and inverted Hg(Cd)Te quantum wells in proximity to conventional superconductors.

In-Plane Zeeman-Field-Induced Majorana Corner and Hinge Modes in an s-Wave Superconductor Heterostructure.

Second-order topological superconductors host Majorana corner and hinge modes in contrast to conventional edge and surface modes in two and three dimensions. However, the realization of such second-order corner modes usually demands unconventional superconducting pairing or complicated junctions or layered structures. Here we show that Majorana corner modes could be realized using a 2D quantum spin Hall insulator in proximity contact with an s-wave superconductor and subject to an in-plane Zeeman field. Beyond a critical value, the in-plane Zeeman field induces opposite effective Dirac masses between adjacent boundaries, leading to one Majorana mode at each corner. A similar paradigm also applies to 3D topological insulators with the emergence of Majorana hinge states. Avoiding complex superconductor pairing and material structure, our scheme provides an experimentally realistic platform for implementing Majorana corner and hinge states.

Second-order bulk-boundary correspondence in rotationally symmetric topological superconductors from stacked Dirac Hamiltonians

Two-dimensional second-order topological superconductors host zero-dimensional Majorana bound states at their boundaries. In this work, focusing on rotation-invariant crystalline topological superconductors, we establish a bulk-boundary correspondence linking the presence of such Majorana bound states to bulk topological invariants introduced by Benalcazar et al. We thus establish when a topological crystalline superconductor protected by rotational symmetry displays second-order topological superconductivity. Our approach is based on stacked Dirac Hamiltonians, using which we relate transitions between topological phases to the transformation properties between adjacent gapped boundaries. We find that in addition to the bulk rotational invariants, the presence of Majorana boundary bound states in a given geometry depends on the interplay between weak topological invariants and the location of the rotation center relative to the lattice. We provide numerical examples for our predictions and discuss possible extensions of our approach.

Magnetic flux periodicity in second order topological superconductors

The magnetic flux periodicity of $\frac{hc}{2e}$ is a well known manifestation of Cooper pairing in typical s-wave superconductors. In this paper we theoretically show that the flux periodicity of a two-dimensional second-order topological superconductor, which features zero-energy Majorana modes localized at the corners of the sample, is $\frac{hc}{e}$ instead. We further show that the periodicity changes back to $\frac{hc}{2e}$ at the transition to a topologically trivial superconductor, where the Majorana modes hybridize with the bulk states, demonstrating that the doubling of periodicity is a manifestation of the non-trivial topology of the state.

Higher-order topological superconductivity of spin-polarized fermions

We study the superconductivity of spin-polarized electrons in centrosymmetric ferromagnetic metals. Due to the spin-polarization and the Fermi statistics of electrons, the superconducting pairing function naturally has odd parity. According to the parity formula proposed by Fu, Berg, and Sato, odd-parity pairing leads to conventional first-order topological superconductivity when a normal metal has an odd number of Fermi surfaces. Here, we derive generalized parity formulae for the topological invariants characterizing higher-order topology of centrosymmetric superconductors. Based on the formulae, we systematically classify all possible band structures of ferromagnetic metals that can induce inversion-protected higher-order topological superconductivity. Among them, doped ferromagnetic nodal semimetals are identified as the most promising normal state platform for higher-order topological superconductivity. In two dimensions, we show that odd-parity pairing of doped Dirac semimetals induces a second-order topological superconductor. In three dimensions, odd-parity pairing of doped nodal line semimetals generates a nodal line topological superconductor with monopole charges. On the other hand, odd-parity pairing of doped monopole nodal line semimetals induces a three-dimensional third-order topological superconductor. Our theory shows that the combination of superconductivity and ferromagnetic nodal semimetals opens up a new avenue for future topological quantum computations using Majorana zero modes.

Majorana corner flat bands in two-dimensional second-order topological superconductors

In this paper we find that confining a second-order topological superconductor with a harmonic potential leads to a proliferation of Majorana corner modes. As a consequence, this results in the formation of Majorana corner flat bands which have a fundamentally different origin from that of the conventional mechanism. This is due to the fact that they arise solely from the one-dimensional gapped boundary states of the hybrid system that become gapless without the bulk gap closing under the increase of the trapping potential magnitude. The Majorana corner states are found to be robust against the strength of the harmonic trap and the transition from Majorana corner states to Majorana flat bands is merely a smooth crossover. As a harmonic trap can potentially be realized in heterostructures, this proposal paves a way to observe these Majorana corner flat bands in an experimental context.

Measurement-only quantum computation with Floquet Majorana corner modes

Majorana modes, typically arising at the edges of one-dimensional topological superconductors, are considered to be a promising candidate for encoding nonlocal qubits in fault-tolerant quantum computation. Here we propose to exploit the two-dimensional geometry of Majorana corner modes in second-order topological superconductors to realize measurement-only quantum computation. In particular, a periodically modulated topological superconducting system may host eight Majorana corner modes, through which two logical qubits as well as one ancilla qubit can be constructed and topologically protected gate operations can be implemented by a series of Majorana parity measurements. The measurements are achieved via Mach-Zehnder type interference in the conductance of a semiconductor based topological superconductor. Our proposal represents a scenario in which single- and two-qubit gate operations can be carried out in a minimal setup that may provide a natural building block for Majorana-based qubit architectures.

Detection of second-order topological superconductors by Josephson junctions

This work uncovers the role of chemical potential in a second order topological superconductor. It results in a zero-$\ensuremath{\pi}$ transition in a Josephson junction as a function of the chemical potential. Additionally, the authors propose a novel platform for creating and manipulating Majorana bound states in a fully electric way.

Majorana corner and hinge modes in second-order topological insulator/superconductor heterostructures

As platforms of Majorana modes, topological insulator (quantum anomalous Hall insulator)/superconductor (SC) heterostructures have attracted tremendous attention over the past decade. Here we substitute the topological insulator by its higher-order counterparts. Concretely, we consider second-order topological insulators (SOTIs) without time-reversal symmetry and investigate SOTI/SC heterostructures in both two and three dimensions. Remarkably, we find that such novel heterostructures provide natural realizations of second-order topological superconductors (SOTSCs) which host Majorana corner modes in two dimensions and chiral Majorana hinge modes in three dimensions. As here the realization of SOTSCs requires neither special pairings nor magnetic fields, such SOTI/SC heterostructures are outstanding platforms of Majorana modes and may have wide applications in future.

Lattice-Symmetry-Assisted Second-Order Topological Superconductors and Majorana Patterns.

We propose a realization of the lattice-symmetry-assisted second-order topological superconductors with corner Majorana zero modes (MZM) based on two-dimensional topological insulators (2DTI). The lattice symmetry can naturally lead to the anisotropic coupling of edge states along different directions to the in-plane magnetic field and conventional s-wave pairings, thus leading to a single MZM located at the corners for various lattice patterns. In particular, we focus on the 2DTI with D_{3d} lattice symmetry and found different types of gap opening for the edge states along the armchair and zigzag edges in a broad range of parameters. As a consequence, a single MZM exists at the corner between the zigzag and armchair edges, and is robust against weakly broken lattice symmetry. We propose to realize such corner MZMs in a variety of polygon patterns, such as triangles and quadrilaterals. We further show their potentials in building the Majorana network through constructing the Majorana Y junction under an in-plane magnetic field.

Phase-tunable second-order topological superconductor

Two-dimensional second-order topological superconductors (SOTSCs) have gapped bulk and edge states, with zero-energy Majorana bound states localized at corners. Motivated by recent advances in Majorana nanowire experiments, we propose to realize a tunable SOTSC as a two-dimensional nanowire array. We show that the coupling between the Majorana modes of adjacent wires can be controlled by phase-biasing the device, allowing to access a variety of topological phases. We characterize the system using scattering theory, which provides access to its transport properties and its topological invariants. The setup is robust against disorder, both in the nanowires themselves and in the Josephson junctions formed between adjacent wires. Further, we identify a parameter regime in which an initially trivial system is rendered topological upon adding disorder, providing an example of a second-order topological Anderson phase.

Second-Order Topological Superconductors with Mixed Pairing.

We show that a two-dimensional semiconductor with Rashba spin-orbit coupling could be driven into the second-order topological superconducting phase when a mixed-pairing state is introduced. The superconducting order we consider involves only even-parity components and meanwhile breaks time-reversal symmetry. As a result, each corner of a square-shaped Rashba semiconductor would host one single Majorana zero mode in the second-order nontrivial phase. Starting from edge physics, we are able to determine the phase boundaries accurately. A simple criterion for the second-order phase is further established, which concerns the relative position between Fermi surfaces and nodal points of the superconducting order parameter. In the end, we propose two setups that may bring this mixed-pairing state into the Rashba semiconductor, followed by a brief discussion on the experimental feasibility of the two platforms.

Weak-pairing higher order topological superconductors

Conventional topological superconductors are fully gapped in the bulk but host gapless Majorana modes on their boundaries. We instead focus on a new class of superconductors, second-order topological superconductors, that have gapped, topological surfaces and gapless Majorana modes instead on lower-dimensional boundaries, i.e., corners of a two-dimensional system or hinges for a three-dimensional system. Here we propose two general scenarios in which second-order topological superconductivity can be realized spontaneously with weak-pairing instabilities. First, we show that $p_x+ip_y$-wave pairing in a (doped) Dirac semimetal in two dimensions with four mirror symmetric Dirac nodes realizes second-order topological superconductivity. Second, we show that $p+id$ pairing on an ordinary spin-degenerate Fermi sruface realizes second-order topological superconductivity as well. In the latter case we find that the topological invariants describing the system can be written using simple formulae involving only the low-energy properties of the Fermi surfaces and superconducting pairing. In both cases we show that these exotic superconducting states can be intrinsically realized in a metallic system with electronic interactions. For the latter case we also show it can be induced by proximity effect in a heterostructure of cuprate and topological superconductors.

Second-order topological insulators and superconductors with an order-two crystalline symmetry

Second-order topological insulators and superconductors have a gapped excitation spectrum in bulk and along boundaries, but protected zero modes at corners of a two-dimensional crystal or protected gapless modes at hinges of a three-dimensional crystal. A second-order topological phase can be induced by the presence of a bulk crystalline symmetry. Building on Shiozaki and Sato's complete classification of bulk crystalline phases with an order-two crystalline symmetry [Phys.\ Rev.\ B {\bf 90}, 165114 (2014)], such as mirror reflection, twofold rotation, or inversion symmetry, we classify all corresponding second-order topological insulators and superconductors. The classification also includes antiunitary symmetries and antisymmetries.

Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields

A two-dimensional second-order topological superconductor exhibits a finite gap in both bulk and edges, with the nontrivial topology manifesting itself through Majorana zero modes localized at the corners, i.e., Majorana corner states. We investigate a time-reversal-invariant topological superconductor in two dimension and demonstrate that an in-plane magnetic field could transform it into a second-order topological superconductor. A detailed analysis reveals that the magnetic field gives rise to mass terms which take distinct values among the edges, and Majorana corner states naturally emerge at the intersection of two adjacent edges with opposite masses. With the rotation of the magnetic field, Majorana corner states localized around the boundary may hop from one corner to a neighboring one and eventually make a full circle around the system when the field rotates by $2\pi$. In the end we briefly discuss physical realizations of this system.