Topological phases of fermions in one dimension

Abstract

In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of 1D systems. We focus on the TR-invariant Majorana chain (BDI symmetry class). While the band classification yields an integer topological index $k$, it is known that phases characterized by values of $k$ in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half-chains. We generalize these results to the classification of all one dimensional gapped phases of fermionic systems with possible anti-unitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.

14 pages, 3 figures, v2: references added

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• Symmetry protected weak topological phases in a superlattice
• Piercing the rainbow: entanglement on an inhomogeneous spin chain with a defect
• Conformal Field Theories generated by Chern Insulators under Quantum Decoherence
• Transmission through a potential barrier in Luttinger liquids with a topological spin gap
• One-dimensional non-interacting topological insulators with chiral symmetry
• Topological d-wave superconductivity in two dimensions
• Bosonization for fermions and parafermions
• Finite Size Scaling of Topological Entanglement Entropy
• Simulation of the multiphase configuration and phase transitions with quantum walks utilizing a step-dependent coin
• Boundary-Induced Topological and Mid-Gap States in Charge Conserving One-Dimensional Superconductors
• Classification of the quantum chaos in colored Sachdev-Ye-Kitaev models
• Symmetry protected phases in inhomogeneous spin chains
• Boundary supersymmetry of 1+1 d fermionic SPT phases
• Interacting topological mirror excitonic insulator in one dimension
• Exact Solution to Haldane-BCS-Hubbard Model Along the Symmetric Lines: Interaction Induced Topological Phase Transition
• Definition and Classification of Fermi Surface Anomalies
• Majorana orthogonal transformation and Majorana zero modes in free fermionic systems
• Classification of symmetry-protected topological many-body localized phases in one dimension
• Toward simulation of topological phenomenas with one-, two- and three-dimensional quantum walks
• Generic "Unnecessary" Quantum Critical Points with Minimal Degrees of Freedom
• Coulomb Blockade in Fractional Topological Superconductors
• Homogeneous and domain wall topological Haldane conductors with dressed Rydberg atoms
• Nonlocal Entanglement of 1D Thermal States Induced by Fermion Exchange Statistics
• Dyonic zero-energy modes
• Renormalization group approach to symmetry protected topological phases
• Topological phase transitions in random Kitaev $\alpha$-chains
• Entanglement sum rules in exactly solvable models
• Symmetry-protected topological phase transition in one-dimensional Kondo lattice and its realization with ultracold atoms
• Gauging anomalous unitary operators
• Interacting Topological Insulators with Synthetic Dimensions
• Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases
• Eight-vertex criticality in the interacting Kitaev chain
• "Haldane" phases with ultracold fermionic atoms in double-well optical lattices
• Unconventional Majorana fermions on the surface of topological superconductors protected by rotational symmetry
• Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension
• Topological phases of parafermionic chains with symmetries
• Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: a Majorana-fermion approach
• Topological Charge/spin density Wave in InAs/GaSb Quantum Wells under an In-plane Magnetic Field
• Simulating Majorana zero modes on a noisy quantum processor
• Interacting Valley Chern Insulator and its Topological Imprint on Moiré Superconductors
• Statistically induced topological phase transitions in a one-dimensional superlattice anyon-Hubbard model
• Correlation-Assisted Quantized Charge Pumping
• Majorana end states in an interacting quantum wire
• End-to-end correlations in the Kitaev chain
• Edge theories of 2D fermionic symmetry protected topological phases protected by unitary Abelian symmetries
• Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi-Point Strange Correlators
• Quantum phases of a one-dimensional Majorana-Bose-Hubbard model
• Quasiparticle excitations in a one-dimensional interacting topological insulator: Application for dopant-based quantum simulation
• Higgs Condensates are Symmetry-Protected Topological Phases: II. $U(1)$ Gauge Theory and Superconductors
• Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy
• Diagrammatic State Sums for 2D Pin-Minus TQFTs
• Decorated defect condensate, a window to unconventional quantum phase transitions in Weyl semimetals
• Gutzwiller Approach for Elementary Excitations in $S=1$ Antiferromagnetic Chains
• Boundary conditions and anomalies of conformal field theories in 1+1 dimensions
• Comparative study of the density matrix embedding theory for the Hubbard models
• Rigorous Index Theory for One-Dimensional Interacting Topological Insulators
• Complexity is not Enough for Randomness
• Interacting Edge States of Fermionic Symmetry-Protected Topological Phases in Two Dimensions
• Transport signatures of interacting fermions in quasi-one-dimensional topological superconductors
• Topological Quantum Critical Points in Strong Coupling limits: Global Symmetries and Strongly Interacting Majorana Fermions
• Phase diagram and sweep dynamics of a one-dimensional generalized cluster model
• A many-body Fredholm index for ground state spaces and Abelian anyons
• Universal finite size scaling around tricriticality between topologically ordered, SPT, and trivial phases
• Superuniversality from disorder at two-dimensional topological phase transitions
• Dynamical topological excitations in parafermion chains
• Gapping fragile topological bands by interactions
• Exotic Invertible Phases with Higher-Group Symmetries
• Transport properties of an interacting Majorana chain
• Enforced symmetry breaking by invertible topological order
• Many-body entanglement and topology from uncertainties and measurement-induced modes
• Entropic topological invariant for a gapped one-dimensional system
• Disconnected entanglement entropy as a marker of edge modes in a periodically driven Kitaev chain
• Quantum Error-Detection at Low Energies
• CT or P Problem and Symmetric Gapped Fermion Solution
• Interaction driven topological phase transitions of hardcore bosons on a two-leg ladder
• On ${\mathbb{Z}}_2$-indices for ground states of fermionic chains
• Symmetry classes, many-body zero modes, and supersymmetry in the complex Sachdev-Ye-Kitaev model
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• Theory of quantum kagome ice and vison zero modes
• Transport across a system with three p-wave superconducting wires: effects of Majorana modes and interactions
• Theory of topological spin Josephson junctions
• 4D beyond-cohomology topological phase protected by $C_2$ symmetry and its boundary theories
• Practical new platform for interaction-enabled topological phases
• Entanglement Chern number for three-dimensional topological insulators: Characterization by Weyl points of entanglement Hamiltonians
• Entanglement in non-critical inhomogeneous quantum chains
• Controllable simulation of topological phases and edge states with quantum walk
• Geometric Approach to 3D Interfaces at Strong Coupling
• Frustration-free Hamiltonians supporting Majorana zero edge modes
• Intercomponent entanglement entropy and spectrum in binary Bose-Einstein condensates
• Supersymmetric boundaries of one-dimensional phases of fermions beyond symmetry-protected topological states
• Specific heat of 2D interacting Majorana fermions from holography
• Trialities of minimally supersymmetric 2d gauge theories
• Non-local Order Parameters and Quantum Entanglement for Fermionic Topological Field Theories
• Anomalous domain wall condensation in a modified Ising chain
• Unsupervised interpretable learning of topological indices invariant under permutations of atomic bands
• Performance of the rigorous renormalization group for first order phase transitions and topological phases
• Topological phase transition and the effect of Hubbard interaction on the one-dimensional topological Kondo insulator
• Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle
• Supersymmetric Free Fermions and Bosons: Locality, Symmetry and Topology
• Exactly solvable lattice models for interacting electronic insulators in two dimensions
• Topological Kondo effect with spinful Majorana fermions
• Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations
• Fermionic Symmetry Protected Topological Phase Induced by Interactions
• Weakly interacting one-dimensional topological insulators: a bosonization approach
• Fusion of one-dimensional gapped phases and their domain walls
• Unveiling Correlated Two-dimensional Topological Insulators through Fermionic Tensor Network States -- Classification, Edge Theories and Variational Wavefunctions
• Critical properties of the Majorana chain with competing interactions
• A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation
• Detection of gapped phases of a 1D spin chain with onsite and spatial symmetries
• Entanglement of a alternating bipartition in spin chains: Relation with classical integrable models
• Dynamical edge modes in Maxwell theory from a BRST perspective, with an application to the Casimir energy
• Role of anomalous symmetry in 0-$\pi$ qubits
• Lattice construction of exotic invertible topological phases
• Exact correlations in topological quantum chains
• Even and odd normalized zero modes in random interacting Majorana models respecting the Parity $P$ and the Time-Reversal-Symmetry $T$
• Topological properties of magnetically ordered heavy-fermion systems in the presence of mirror symmetry
• Interplay between intrinsic and emergent topological protection on interacting helical modes
• Interacting topological phases in thin films of topological mirror Kondo insulators
• Topological end states in a one-dimensional spatially modulated interaction spinless fermion model
• Quaternary Jordan-Wigner mapping and topological extended-kink phase in the interacting Kitaev ring
• Bound states of three fermions forming symmetry-protected topological phases
• Interactions and the Theta Term in One-Dimensional Gapped Systems
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• C-P-T Fractionalization
• Platform for controllable Majorana zero modes using superconductor/ferromagnet heterostructures
• A Bosonic Analog of a Topological Dirac Semi-Metal: Effective Theory, Neighboring Phases, and Wire Construction
• Renormalization view on resonance proliferation between many-body localized phases
• Gauging Defects in Quantum Spin Systems
• Quantum Chaos, Thermodynamics and Black Hole Microstates in the mass deformed SYK model
• A search for correlation-induced adiabatic paths between distinct topological insulators
• Fermionic symmetry-protected topological state in strained graphene
• Solvable lattice model for (2+1)D bosonic topological insulator
• Unwinding of a one-dimensional topological superconductor
• Topology and Spectrum in Measurement-Induced Phase Transitions
• Interacting Majorana modes at surfaces of noncentrosymmetric superconductors
• Building Symmetry Enriched Topological Phases from a Bipartite Lattice Construction and Anyon Condensation
• Interacting Chern Insulator in Infinite Spatial Dimensions
• Stability against contact interactions of a topological superconductor in two-dimensional space protected by time-reversal and reflection symmetries
• Topological quantum phase transitions in a Majorana chain with spatial modulation
• Commuting projector models for (3+1)d topological superconductors via string net of (1+1)d topological superconductors
• Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems
• Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions
• Classifying parafermionic gapped phases using matrix product states
• G-Extensions of Quantum Group Categories and Functorial SPT
• Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. II. Topological classification
• Reduction of $\mathbb{Z}$ classification of a two-dimensional weak topological insulator - real-space DMFT study -
• Quantum-critical dynamics of a Josephson junction at the topological transition
• Random Quantum Circuits with Time-Reversal Symmetry
• Complete characterization of spin chains with two Ising symmetries
• Topological Superconductivity in Doped Symmetry Protected Topological State
• Chiral topological states in Bose-Fermi mixtures
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Majorana zero modes in a quantum wire platform without Rashba spin-orbit coupling
• A geometrical crossover in excited states of two-electron quantum dots in a magnetic field
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
• Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
• Topological Dipole Insulator
• Generalized Li-Haldane Correspondence in Critical Dirac-Fermion Systems
• Statistic behaviors of gauge-invariance-dominated 1D chiral current random model
• Quantum Restored Symmetry Protected Topological Phases
• Enriched classification of parafermionic gapped phases with time reversal symmetry
• A Classifying Space for Phases of Matrix Product States
• 1D Error Correcting Code for Majorana Qubits
• Construction of Many-Body-Localized Models where all the eigenstates are Matrix-Product-States
• Variational Tensor Wavefunctions for the Interacting Quantum Spin Hall Phase
• Hierarchical analytical approach to universal spectral correlations in Brownian Quantum Chaos
• Lowest energy states of an $O(N)$ fermionic chain
• Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories
• Quantifying robustness and locality of Majorana bound states in interacting systems
• Topological Degeneracy Induced Flat Bands in two-Dimensional Holed Systems
• Les Houches Lecture Notes on Tensor Networks
• Inter-species topological phases via a dynamical gauge field
• Computational Many-Body Physics via ${\mathcal{M}}_{2^q}$ Algebra
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Exploring two dimensional ${\mathbb{Z}}_2$ invariant phases with time reversal symmetry and their transitions with topological operations
• Quantum Degenerate Majorana Surface Zero Modes in Two-Dimensional Space
• Overlap of parafermionic zero modes at a finite distance
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients
• Intertwined order of generalized global symmetries
• Threefold Way for Typical Entanglement
• Characterizing the parent Hamiltonian for a complete set of orthogonal wave functions: An inverse quantum problem
• (2+1)D topological phases with RT symmetry: many-body invariant, classification, and higher order edge modes
• Emergence of a new symmetry class for Bogoliubov-de Gennes (BdG) Hamiltonians: Expanding ten-fold symmetry classes
• Tunable Magnetic Order in Chiral Coupled Spin Chains
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Impurity screening by defects in (1+1)$d$ quantum critical systems
• Reentrant topology and reverse pumping in a quasiperiodic flux ladder
• Out-of-time-order correlation and detection of phase structure in Floquet transverse Ising spin system
• Symmetry Reduction and Boundary Modes for Fe-Chains on an s-wave Superconductor
• Reflection-Symmetry Protected Antiferromagnetic Topological Insulator in Three-Dimensional Heavy-Fermion Systems
• Transport in extended Kitaev chain with time reversal symmetry breaking and long-range interaction
• Generalized symmetry enriched criticality in (3+1)d
• Correspondence of topological classification between quantum graph extra dimension and topological matter
• Web of dualities on non-orientable surfaces
• One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems
• Equivalence class of Emergent Single Weyl fermion lattice models in 3 dimensions: gapless superconductors and superfluids versus chiral fermions
• Non-local order parameters for fermion chains via the partial transpose
• Adiabatic preparation of a number-conserving atomic Majorana phase
• Majorana edge modes in isolated wires
• Surface topological quantum criticality: Conformal manifolds and Discrete Strong Coupling Fixed Points
• Entanglement entropy scaling laws from fluctuations of non-conserved quantities
• Rényi entropy of the permutationally invariant part of the ground state across a quantum phase transition
• Towards a Topological Quantum Chemistry description of correlated systems: the case of the Hubbard diamond chain
• Quantum Avalanches in ${\mathbb{Z}}_2$-preserving Interacting Ising Majorana Chain
• Emergent Haldane phase in an alternating bond ${\mathbb{Z}}_3$ parafermion chain
• Spin order, spin excitations, and RIXS spectra of spin-1/2 tetramer chains
• Geometric Approaches to Quantum Fields and Strings at Strong Couplings