Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D

Abstract

Topological superconductors in one spatial dimension exhibiting a single Majorana bound state at each end are distinguished from trivial gapped systems by a Z_2 topological invariant. Originally, this invariant was calculated by Kitaev in terms of the Pfaffian of the Majorana representation of the Hamiltonian: The sign of this Pfaffian divides the set of all gapped quadratic forms of Majorana fermions into two inequivalent classes. In the more familiar Bogoliubov de Gennes mean field description of superconductivity, an emergent particle hole symmetry gives rise to a quantized Zak-Berry phase the value of which is also a topological invariant. In this work, we explicitly show the equivalence of these two formulations by relating both of them to the phase winding of the transformation matrix that brings the Majorana representation matrix of the Hamiltonian into its Jordan normal form.

5 pages; v2: minor changes, ref. added

References in corpus (8)

• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Classification of topological insulators and superconductors in three spatial dimensions
• Evidence of Majorana fermions in an Al - InAs nanowire topological superconductor
• Observation of Majorana Fermions in a Nb-InSb Nanowire-Nb Hybrid Quantum Device
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• On the $Z_2$ classification of Quantum Spin Hall Models
• From the adiabatic theorem of quantum mechanics to topological states of matter

Cited by in corpus (56)

• Classification of topological quantum matter with symmetries
• Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor
• Topological superconductivity on the surface of Fe-based superconductors
• An anomalous higher-order topological insulator
• Topological Superconductivity induced by Ferromagnetic Metal Chains
• Majorana nanowires for topological quantum computation
• Extended Kitaev chain with longer-range hopping and pairing
• Topology of one dimensional quantum systems out of equilibrium
• Classification of topological insulators and superconductors out of equilibrium
• Real space mapping of topological invariants using artificial neural networks
• Quasi 1D topological nodal vortex line phase in doped superconducting 3D Dirac Semimetals
• Topological phase transition in coupled rock-paper-scissor cycles
• Signatures of topological phase transitions in Josephson current-phase discontinuities
• Topological invariant for generic 1D time reversal symmetric superconductors in class DIII
• Controlling Majorana states in topologically inhomogeneous superconductors
• Second-Order Bulk-Boundary Correspondence in Rotationally Symmetric Topological Superconductors from Stacked Dirac Hamiltonians
• Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires
• Exact treatment of magnetism-driven ferroelectricity in the one-dimensional compass model
• Exact ground states for interacting Kitaev chains
• Topological polarons, quasiparticle invariants and their detection in 1D symmetry-protected phases
• Topological Vortex Phase Transitions in Iron-Based Superconductors
• Embedded Topological Insulators
• Topological polarization, dual invariants, and surface flat band in crystalline insulators
• Nodal-line semimetals from Weyl superlattices
• Topological invariants for phase transition points of one-dimensional ${\mathbb{Z}}_2$ topological systems
• Free and Interacting Short-Range Entangled Phases of Fermions: Beyond the Ten-Fold Way
• Strain induced topological phase transition at zigzag edges of monolayer transition-metal dichalcogenides
• Majorana zero modes in gate-defined germanium hole nanowires
• Atomic line defects and topological superconductivity in unconventional superconductors
• Synthesizing Majorana zero-energy modes in a periodically gated quantum wire
• Tomography of zero-energy end modes in topological superconducting wires
• Phase space methods for Majorana fermions
• Phase diagram of a model for topological superconducting wires
• Unconventional quantum phase transitions in a one-dimensional Lieb-Schultz-Mattis system
• Intertwined Order in Fractional Chern Insulators from Finite-Momentum Pairing of Composite Fermions
• Pfaffian formula for fermion parity fluctuations in a superconductor and application to Majorana fusion detection
• Majorana fermions in topological insulator nanowires: from single superconducting nanowires to Josephson junctions
• Transport features of a topological superconducting nanowire with a quantum dot: conductance and noise
• Interaction-induced topological superconductivity in antiferromagnet-superconductor junctions
• Topological indicators for systems with open boundaries: Application to the Kitaev wire
• An Interplay of Topology and Quantized Geometric Phase for two Different Symmetry-Class Hamiltonians
• Unsupervised interpretable learning of topological indices invariant under permutations of atomic bands
• Majorana modes in striped two-dimensional inhomogeneous topological superconductors
• Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii-Moriya Interaction
• Lifshitz transitions and Weyl semimetals from a topological superconductor with supercurrent flow
• Fragile topology in nodal-line semimetal superconductors
• One-dimensional ${\mathbb{Z}}_4$ topological superconductor
• Quantized frequency-domain polarization of driven phases of matter
• Perfectly localized Majorana corner modes in fermionic lattices
• Chiral Majorana Edge Modes and Vortex Majorana Zero Modes in Superconducting Antiferromagnetic Topological Insulator
• Topological Classification of One-Dimensional Chiral Symmetric Interfaces
• Extended edge modes and disorder preservation of a symmetry-protected topological phase out of equilibrium
• Origin of Robust ${\mathbb{Z}}_2$ Topological Phases in Stacked Hermitian Systems: Non-Hermitian Level Repulsion
• Tunable zero modes and quantum interferences in flat-band topological insulators
• Charge current and phase diagram of the disordered open longer-range Kitaev chain
• Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors