Squaring the fermion: The threefold way and the fate of zero modes

Abstract

We investigate topological properties and classification of mean-field theories of stable bosonic systems. Of the three standard classifying symmetries, only time-reversal represents a real symmetry of the many-boson system, while the other two, particle-hole and chiral, are simply constraints that manifest as symmetries of the effective single-particle problem. For gapped systems in arbitrary space dimension we establish three fundamental no-go theorems that prove the absence of: parity switches, symmetry-protected-topological quantum phases, and localized bosonic zero modes under open boundary conditions. We then introduce a squaring, kernel-preserving map connecting non-interacting Hermitian theories of fermions and stable boson systems, which serves as a playground to reveal the role of topology in bosonic phases and their localized midgap boundary modes. Finally, we determine the symmetry classes inherited from the fermionic tenfold-way classification, unveiling an elegant threefold-way topological classification of non-interacting bosons. We illustrate our main findings in one- and two-dimensional bosonic lattice and field-theory models.

arXiv comment26+5 pages, 3 figures, 9 tables

References in corpus (30)

• Unpaired Majorana fermions in quantum wires
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Classification of topological insulators and superconductors in three spatial dimensions
• 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
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Search for Majorana fermions in superconductors
• Symmetry and Topology in Non-Hermitian Physics
• Topological chiral magnonic edge mode in a magnonic crystal
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Random Matrix Theory and Chiral Symmetry in QCD
• Majorana modes in time-reversal invariant s-wave topological superconductors
• Topological symmetry classes for non-Hermitian models and connections to the bosonic Bogoliubov-de Gennes equation
• Many-body characterization of topological superconductivity: The Richardson-Gaudin-Kitaev chain
• A generalization of Bloch's theorem for arbitrary boundary conditions: Theory
• A Classification of Non-Hermitian Random Matrices
• Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians
• Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions
• Bosonic quadratic Hamiltonians
• Topological edge states for disordered bosonic systems
• Majorana Flat Bands in s-Wave Gapless Topological Superconductors
• Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure
• Multiband s-wave topological superconductors: role of dimensionality and magnetic field response
• What is a particle-conserving Topological Superfluid?
• Exact solution of corner-modified banded block-Toeplitz eigensystems
• Equivalence of Topological Insulators and Superconductors
• Dirac cones and mass terms in bosonic spectra
• Dirac Hamiltonians for bosonic spectra
• Integrable model of a $p$-wave bosonic superfluid

Cited by in corpus (25)

• Topological Magnons: A Review
• Catalog of Unconventional Magnons in Collinear Magnets
• Spin-Space Groups and Magnon Band Topology
• Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics
• Deconstructing effective non-Hermitian dynamics in quadratic bosonic Hamiltonians
• Symmetry and topological classification of Floquet non-Hermitian systems
• Topological magnetic excitations
• Why the first magic-angle is different from others in twisted graphene bilayers: interlayer currents, kinetic and confinement energy and wavefunction localization
• Topological zero modes and edge symmetries of metastable Markovian bosonic systems
• Bulk-boundary correspondence and singularity-filling in long-range free-fermion chains
• Identifying, and constructing, complex magnon band topology
• A simple approach to characterizing band topology in bosonic pairing Hamiltonians
• The Interplay of Finite and Infinite Size Stability in Quadratic Bosonic Lindbladians
• Supersymmetry on the lattice: Geometry, Topology, and Flat Bands
• Zero-frequency chiral magnonic edge states protected by non-equilibrium topology
• Krein-unitary Schrieffer-Wolff transformation and band touchings in bosonic Bogoliubov-de Gennes and other Krein-Hermitian Hamiltonians
• Quantum geometry of bosonic Bogoliubov quasiparticles
• Quantum geometry and magnon Hall transport in an altermagnet
• Supersymmetric Free Fermions and Bosons: Locality, Symmetry and Topology
• Topological classifications of quadratic bosonic excitations in closed and open systems with examples
• Quantum Weyl-Heisenberg antiferromagnet
• Topological bosonic Bogoliubov excitations with sublattice symmetry
• Bosonic Holes in Quadratic Bosonic Systems
• Bloch and Bethe ansatze for the Harper model: A butterfly with a boundary
• Proof of bulk-edge correspondence for band topology by Toeplitz algebra