On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains

Abstract

For parity-conserving fermionic chains, we review how to associate ${\mathbb{Z}}_2$-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the ${\mathbb{Z}}_2$-valued spectral flow provides a topological obstruction for two systems to have the same ${\mathbb{Z}}_2$-index. A rudimentary definition of a ${\mathbb{Z}}_2$-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

arXiv commentv2: 61 pages, many revisions. To appear in Reviews in Mathematical Physics

References in corpus (34)

• Anyons in an exactly solved model and beyond
• Unpaired Majorana fermions in quantum wires
• An Area Law for One Dimensional Quantum Systems
• Topological phases of fermions in one dimension
• Spectral Gap and Exponential Decay of Correlations
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Stability of Frustration-Free Hamiltonians
• Exact ground states and topological order in interacting Kitaev/Majorana chains
• Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Quasi-Locality Bounds for Quantum Lattice Systems. Part I. Lieb-Robinson Bounds, Quasi-Local Maps, and Spectral Flow Automorphisms
• Quantization of Hall Conductance For Interacting Electrons on a Torus
• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
• The adiabatic theorem and linear response theory for extended quantum systems
• Lieb-Schultz-Mattis type theorems for quantum spin chains without continuous symmetry
• Minimalist approach to the classification of symmetry protected topological phases
• Universality of the Hall conductivity in interacting electron systems
• Fermionic projected entangled-pair states and topological phases
• Index pairings in presence of symmetries with applications to topological insulators
• Exact zero modes in twisted Kitaev chains
• Spectral flows associated to flux tubes
• Adiabatic currents for interacting electrons on a lattice
• Spin Topological Field Theory and Fermionic Matrix Product States
• On gapped phases with a continuous symmetry and boundary operators
• Fermionic Matrix Product States and One-Dimensional Short-Range Entangled Phases with Anti-Unitary Symmetries
• On the stability of charges in infinite quantum spin systems
• Stability of Gapped Ground State Phases of Spins and Fermions in One Dimension
• $C^1$-classification of gapped parent Hamiltonians of quantum spin chains
• Continuity of the spectrum of a field of self-adjoint operators
• A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III
• Automorphic equivalence preserves the split property
• The non-commutative topology of two-dimensional dirty superconductors
• Parity flow as ${\mathbb{Z}}_2$-valued spectral flow
• Majorana Fermions and Orthogonal Complex Structures

Cited by in corpus (7)

• The classification of symmetry protected topological phases of one-dimensional fermion systems
• The KO-valued spectral flow for skew-adjoint Fredholm operators
• The split and approximate split property in 2D systems: stability and absence of superselection sectors
• Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems
• Rigorous Index Theory for One-Dimensional Interacting Topological Insulators
• Locally equivalent quasifree states and index theory
• A ${\mathbb{Z}}_2$-Topological Index for Quasi-Free Fermions