Floquet first- and higher-order topological superconductors: A review

Abstract

The current developments in quantum technologies are deeply rooted in quantum computation, with Majorana fermions and their topological protection expected to play a central role. In this context, topological superconductors (TSCs) provide a promising platform for investigating Majorana modes and their potential applications in topological quantum computation. In dimensions $d>1$, the spatial localization of topological Majorana modes can be controlled in higher-order topological superconductors (HOTSCs), extending beyond the conventional framework of first-order topological superconductors (FOTSCs). In this review, we systematically survey the diverse landscape of topological superconductivity, ranging from FOTSCs to third-order topological superconductors (TOTSCs), in which the spatial dimensionality of Majorana zero modes (MZMs) is progressively reduced. We demonstrate how periodic driving protocols enable Floquet engineering of all these phases, even when the corresponding undriven systems are topologically trivial. Furthermore, we investigate various driving schemes and explore the interplay among spin-orbit coupling, magnetic fields, magnetic impurities, and symmetry-breaking mass terms in realizing Floquet higher-order topological superconducting (FHOTSC) phases, characterized by appropriate topological invariants. We also provide an extensive discussion of topological invariants capable of distinguishing conventional MZMs from anomalous Majorana $pi$-modes, which emerge at two distinct quasienergies and have no static counterparts.

arXiv comment28 pages without references and 16 figures- Invited Topical Review in Journal of Physics: Condensed Matter

References in corpus (186)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Non-Abelian Anyons and Topological Quantum Computation
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Unpaired Majorana fermions in quantum wires
• Topological Insulators in Three Dimensions
• Topological Insulators with Inversion Symmetry
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Discovery (theoretical prediction and experimental observation) of a large-gap topological-insulator class with spin-polarized single-Dirac-cone on the surface
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Helical liquids and Majorana bound states in quantum wires
• Classification of topological quantum matter with symmetries
• New directions in the pursuit of Majorana fermions in solid state systems
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Quantum Spin Hall Effect
• Evidence of Majorana fermions in an Al - InAs nanowire topological superconductor
• Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor
• Quantized Electric Multipole Insulators
• A generic new platform for topological quantum computation using semiconductor heterostructures
• Floquet Topological Insulator in Semiconductor Quantum Wells
• Higher-Order Topological Insulators
• Topological superconductors: a review
• Photovoltaic Hall effect in graphene
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Topological characterization of periodically-driven quantum systems
• Atomic quantum gases in periodically driven optical lattices
• Observation of the fractional ac Josephson effect: the signature of Majorana particles
• Majorana Zero Modes and Topological Quantum Computation
• The Magnus expansion and some of its applications
• Transport properties of non-equilibrium systems under the application of light: Photo-induced quantum Hall insulators without Landau levels
• Exponential Protection of Zero Modes in Majorana Islands
• Majorana bound states in a coupled quantum-dot hybrid-nanowire system
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Topolectrical circuit realization of topological corner modes
• Reflection symmetric second-order topological insulators and superconductors
• Observation of a phononic quadrupole topological insulator
• Floquet Engineering of Quantum Materials
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• Acoustic higher-order topological insulator on a Kagome lattice
• Higher-Order Topology in Bismuth
• Anomalous modulation of a zero bias peak in a hybrid nanowire-superconductor device
• Majorana Fermions in Equilibrium and Driven Cold Atom Quantum Wires
• Majorana fermions in chains of magnetic atoms on a superconductor
• Non-Abelian Topological Order in S-Wave Superfluids of Ultracold Fermionic Atoms
• The Quantum-Mechanical Position Operator in Extended Systems
• Topological Superconductivity and Superfluidity
• High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective
• Photo-Thermoelectric Effect at a Graphene Interface Junction
• Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals
• Direct observation of corner states in second-order topological photonic crystal slabs
• Chiral Topological Superconductor From the Quantum Hall State
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Photonic quadrupole topological phases
• Higher-order topological insulators and superconductors protected by inversion symmetry
• Topological Superconductivity and Majorana Fermions in RKKY Systems
• Topological superconducting phase in helical Shiba chains
• Colloquium: Quantum anomalous Hall effect
• Chiral Majorana edge state in a quantum anomalous Hall insulator-superconductor structure
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• $4\pi$-periodic Josephson supercurrent in HgTe-based topological Josephson junctions
• Floquet Spectrum and Transport Through an Irradiated Graphene Ribbon
• Irradiated graphene as a tunable Floquet topological insulator
• Brillouin-Wigner theory for high-frequency expansion in periodically driven systems: Application to Floquet topological insulators
• Higher-Order Topology, Monopole Nodal Lines, and the Origin of Large Fermi Arcs in Transition Metal Dichalcogenides XTe${}_2$ (X=Mo,W)
• Floquet chiral edge states in graphene
• Topological singularities and the general classification of Floquet-Bloch systems
• Light-induced emergent phenomena in 2D materials and topological materials
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Majorana corner modes in a high-temperature platform
• Non-Abelian Topological Orders and Majorana Fermions in Spin-Singlet Superconductors
• Time Reversal Invariant Topological Superconductivity and Majorana Kramers Pairs
• Floquet generation of Majorana end modes and topological invariants
• The anomalous Floquet-Anderson insulator as a non-adiabatic quantized charge pump
• Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
• Robust and clean Majorana zero mode in the vortex core of high-temperature superconductor (Li0.84Fe0.16)OHFeSe
• Higher Order Topological Phases: A General Principle of Construction
• Weak-Pairing Higher Order Topological Superconductors
• Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects
• Optically engineering the topological properties of a spin Hall insulator
• Two-dimensional p-wave superconducting states with magnetic moments on a conventional s-wave superconductor
• Periodic steady state and interference in a periodically driven quantum system
• An anomalous higher-order topological insulator
• Proximity-induced high-temperature superconductivity in topological insulators Bi2Se3 and Bi2Te3
• Material design with the van der Waals stacking of bismuth-halide chains realizing a higher-order topological insulator
• Demonstration of a quantized acoustic octupole topological insulator
• Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields
• Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor
• Second Order Topological Superconductivity in $\pi$-Junction Rashba Layers
• Dimensional hierarchy of higher-order topology in three-dimensional sonic crystals
• Helical Hinge Majoranas in Iron-Based Superconductors
• Majorana states in helical Shiba chains and ladders
• Unconventional topological phase transitions in helical Shiba chains
• Many-Body Electric Multipole Operators in Extended Systems
• Many-Body Order Parameters for Multipoles in Solids
• Coupled-wire construction of static and Floquet second-order topological insulators
• Floquet Higher-Order Topological Insulators with Anomalous Dynamical Polarization
• Evidence of topological Shiba bands in artificial spin chains on superconductors
• Room-temperature quantum spin Hall edge state in a higher-order topological insulator Bi${}_4$Br${}_4$
• Controlled length-dependent interaction of Majorana modes in Yu-Shiba-Rusinov chains
• Second-order Dirac superconductors and magnetic field induced Majorana hinge modes
• In-plane Zeeman field induced Majorana corner and hinge modes in an $s$-wave superconductor heterostructure
• Complex classes of periodically driven topological lattice systems
• Floquet engineering of long-range p-wave superconductivity
• Topological and holonomic quantum computation based on second-order topological superconductors
• Majorana corner and hinge modes in second-order topological insulator-superconductor heterostructures
• Topological superconductivity of spin-3/2 carriers in a three-dimensional doped Luttinger semimetal
• Edge properties of the chiral d-wave superconducting state in doped graphene
• Dynamics of the vacuum state in a periodically driven Rydberg chain
• Shiba impurity bound states as a probe of topological superconductivity and Fermion parity changing quantum phase transitions
• Out of equilibrium higher-order topological insulator: Floquet engineering and quench dynamics
• Dynamical localization in a chain of hard core bosons under a periodic driving
• Evidence for Higher order topology in Bi and Bi${}_{0.92}$Sb${}_{0.08}$
• Spin-polarized Yu-Shiba-Rusinov states in an iron based superconductor
• Higher-order topological superconductors in $\mathcal{P}$-, $\mathcal{T}$-odd quadrupolar Dirac materials
• Hierarchy of higher-order Floquet topological phases in three dimensions
• Majorana fermions in ferromagnetic chains on the surface of bulk spin-orbit coupled $s$-wave superconductors
• Topological classification of quasi-periodically driven quantum systems
• Floquet generation of Second Order Topological Superconductor
• Spin-orbit coupling induced splitting of Yu-Shiba-Rusinov states in antiferromagnetic dimers
• Higher Order Topological Insulator via Periodic Driving
• All-electrically tunable networks of Majorana bound states
• Spiral magnetic order and topological superconductivity in a chain of magnetic adatoms on a two-dimensional superconductor
• Dynamical synchronization transition in interacting electron systems
• Floquet second-order topological insulators in non-Hermitian systems
• Vortex and Surface Phase Transitions in Superconducting Higher-order Topological Insulators
• Controlling in-gap end states by linking nonmagnetic atoms and artificially-constructed spin chains on superconductors
• Quasi-periodically driven quantum systems
• Topological Phases of Inhomogeneous Superconductivity
• Analytic approaches to periodically driven closed quantum systems: Methods and Applications
• Hierarchy of higher-order topological superconductors in three dimensions
• Simple model for second-order topological insulators and loop-nodal semimetals in Transition Metal Dichalcogenides XTe${}_2$ (X=Mo,W)
• Anomalous and normal dislocation modes in Floquet topological insulators
• Yu-Shiba-Rusinov states in real metals
• Majorana corner modes with solitons in an attractive Hubbard-Hofstadter model of cold atom optical lattices
• Flat Majorana bands in 2-d lattices with inhomogeneous magnetic fields: topology and stability
• Detection of second-order topological superconductors by Josephson junctions
• Topological properties of chains of magnetic impurities on a superconducting substrate: Interplay between the Shiba band and ferromagnetic wire limits
• Non-Hermitian higher-order topological superconductors in two-dimension: statics and dynamics
• Kramers pairs of Majorana corner states in a topological insulator bilayer
• Majorana modes in emergent-wire phases of helical and cycloidal magnet-superconductor hybrids
• Maximum group velocity in a one-dimensional model with a sinusoidally varying staggered potential
• Floquet Second Order Topological Superconductor based on Unconventional Pairing
• Yu-Shiba-Rusinov states and topological superconductivity in Ising paired superconductors
• Floquet engineering of low-energy dispersions and dynamical localization in a periodically kicked three-band system
• Skyrmion Control of Majorana States in Planar Josephson Junctions
• Thermo-electric transport properties of Floquet multi-Weyl Semimetals
• Systematic generation of the cascade of anomalous dynamical first and higher-order modes in Floquet topological insulators
• Topological superconductivity from magnetic impurities on monolayer NbSe${}_2$
• Anomalous Floquet Chiral Topological Superconductivity in a Topological Insulator Sandwich Structure
• Topological characterization and stability of Floquet Majorana modes in Rashba nanowire
• Majorana bound state localization and energy oscillations for magnetic impurity chains on conventional superconductors
• Dissipative Floquet Majorana modes in proximity-induced topological superconductors
• Dynamical construction of Quadrupolar and Octupolar topological superconductors
• Disordered Chern insulator with a two step Floquet drive
• Identification of topological superconductivity in magnetic impurity systems using bulk spin-polarization
• Generation of higher-order topological insulators using periodic driving
• Quadrupole spin polarization as signature of second-order topological superconductors
• Prevalence of trivial zero-energy sub-gap states in non-uniform helical spin chains on the surface of superconductors
• Engineering anomalous Floquet Majorana modes and their time evolution in helical Shiba chain
• Exact Delta Kick Cooling, Time-Optimal Control of Scale-Invariant Dynamics, and Shortcuts to Adiabaticity Assisted by Kicks
• Hinge mode dynamics of periodically driven higher-order Weyl semimetals
• Exact results in Floquet coin toss for driven integrable models
• Yu-Shiba-Rusinov states in 2D superconductors with arbitrary Fermi contours
• Magnetic Flux Periodicity in Second Order Topological Superconductors
• Dynamics of fluctuation correlation in periodically driven classical system
• Tailoring phase transition from topological superconductor to trivial superconductor induced by magnetic textures of a spin-chain on a $p$-wave superconductor
• The fate of current, residual energy and entanglement entropy in aperiodic driving of one dimensional Jordan Wigner integrable models
• Periodic and aperiodic dynamics of flat bands in diamond-octagon lattice
• Time evolution of Majorana corner modes in Floquet second-order topological superconductor
• Enhanced Majorana bound states in magnetic chains on superconducting topological insulator edges
• Yu-Shiba-Rusinov States and Ordering of Magnetic Impurities Near the Boundary of a Superconducting Nanowire
• Higher-order topological corner states induced solely by onsite potentials with mirror symmetry
• Majorana zero-modes in a dissipative Rashba nanowire
• Thermodynamics of a Higher-Order Topological Insulator
• Josephson current signature of Floquet Majorana and topological accidental zero modes in altermagnet heterostructures
• Mixed higher-order topology and nodal and nodeless flat band topological phases in a superconducting multiorbital model
• Majorana corner states in an attractive quantum spin Hall insulator with opposite in-plane Zeeman energy at two sublattice sites
• Fano resonances for tilted linear and quadratic band touching dispersions in a harmonically driven potential well
• Mixed-parity octupolar pairing and corner Majorana modes in three dimensions
• Dissipation induced Majarona $0$- and $\pi$-modes in a driven Rashba nanowire
• Extended Haldane model -- a modern gateway to topological insulators
• Topological energy pumping in a quasi-periodically driven four-level system