Gapless topological phases and symmetry-enriched quantum criticality

Abstract

We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants, establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the $1+1d$ case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors. An example of the former is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the latter arises between the gapped Ising and cluster phases: this symmetry-enriched Ising CFT has an edge mode with finite-size splitting $\sim 1/L^{14}$. In addition to such new cases, our formalism unifies various examples previously studied in the literature. Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs. This raises the question of classification, to which we give a partial answer -- including a complete characterization of symmetry-enriched $1+1d$ Ising CFTs. Non-trivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched $2+1d$ CFT without gapped sectors.

arXiv commentwe generalized our framework to 2+1d, and added more examples

References in corpus (104)

• Unpaired Majorana fermions in quantum wires
• "Deconfined" quantum critical points
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• A class of quantum many-body states that can be efficiently simulated
• Topological phases of fermions in one dimension
• Classifying quantum phases using Matrix Product States and PEPS
• Symmetry Protected Topological phases of Quantum Matter
• Braiding statistics approach to symmetry-protected topological phases
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• The Bootstrap Program for Boundary CFT_d
• Detection of Symmetry Protected Topological Phases in 1D
• Conformal Field Theories Near a Boundary in General Dimensions
• Ground State Phase Diagram of S=1 XXZ Chains with Uniaxial Single-Ion-Type Anisotropy
• Kitaev chains with long-range pairing
• Symmetry protected topological phases from decorated domain walls
• One-Dimensional Symmetry Protected Topological Phases and their Transitions
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Bootstrapping the 3d Ising twist defect
• Majorana Zero Modes in 1D Quantum Wires Without Long-Ranged Superconducting Order
• On Orbifolds with Discrete Torsion
• Theory of defects in Abelian topological states
• Boundary and Interface CFTs from the Conformal Bootstrap
• Long-range Ising and Kitaev Models: Phases, Correlations and Edge Modes
• Topological phases with generalized global symmetries
• The N-component Ginzburg-Landau Hamiltonian with cubic anisotropy: a six-loop study
• Energy Momentum Tensor in Conformal Field Theories Near a Boundary
• Topology and edge modes in quantum critical chains
• Intrinsically Gapless Topological Phases
• Gapless Symmetry Protected Topological Order
• Defects in conformal field theory
• Symmetry fractionalization and twist defects
• A Gapless Symmetry-Protected Topological Phase of Fermions in One Dimension
• The phase diagram of the anisotropic spin-2 XXZ model: an infinite system DMRG study
• Anisotropic perturbations in three-dimensional O(N)-symmetric vector models
• Bootstrapping Heisenberg Magnets and their Cubic Instability
• Quantum phase transition between cluster and antiferromagnetic states
• Random Matrix Theory and Entanglement in Quantum Spin Chains
• A number conserving theory for topologically protected degeneracy in one-dimensional fermions
• Topological invariants for gauge theories and symmetry-protected topological phases
• Majorana Edge States in Interacting Two-chain Ladders of Fermions
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Majorana edge states in two atomic wires coupled by pair-hopping
• Many-body characterization of topological superconductivity: The Richardson-Gaudin-Kitaev chain
• Topological Luttinger Liquids from Decorated Domain Walls
• Localized Majorana-like modes in a number conserving setting: An exactly solvable model
• Ising ferromagnet to valence bond solid transition in a one-dimensional spin chain: Analogies to deconfined quantum critical points
• Deconfined quantum critical point in one dimension
• Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions
• Unconventional Surface Critical Behaviors Induced by Quantum Phase Transition from Two-Dimensional Affleck-Kennedy-Lieb-Tasaki Phase to Néel Order
• Line and surface defects for the free scalar field
• The Prediction of a Gapless Topological "Haldane Liquid" Phase in a One-Dimensional Cold Polar Molecular Lattice
• Discrete Torsion
• Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories
• Topological States in a One-Dimensional Fermi Gas with Attractive Interactions
• Bosonic topological phases of matter: bulk-boundary correspondence, SPT invariants and gauging
• Topological states in a microscopic model of interacting fermions
• Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions
• Gapped boundaries, group cohomology and fault-tolerant logical gates
• Topological degeneracy and pairing in a one-dimensional gas of spinless Fermions
• Topological Quantum Liquids with Long-Range Couplings
• Six-loop $\epsilon$ expansion study of three-dimensional $n$-vector model with cubic anisotropy
• The hidden symmetry-breaking picture of symmetry-protected topological order
• Topological phases in gapped edges of fractionalized systems
• Spinning operators and defects in conformal field theory
• Universal Features of Higher-Form Symmetries at Phase Transitions
• Detecting two dimensional symmetry protected topological order in a ground state wave function
• Unconventional Fusion and Braiding of Topological Defects in a Lattice Model
• Phase diagram of the S=1/2 quantum spin chain with bond alternation
• Higher-form symmetry breaking at Ising transitions
• Symmetry protected topological phases of 1D interacting fermions with spin-charge separation
• The phase transitions between $Z_n\times Z_n$ bosonic topological phases in 1+1 D, and a constraint on the central charge for the critical points between bosonic symmetry protected topological phases
• Rokhsar-Kivelson Models of Bosonic Symmetry-Protected Topological States
• Emergent topological properties in interacting one-dimensional systems with spin-orbit coupling
• Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems
• Edge states of the long-range Kitaev chain: an analytical study
• Non-local Order in Elongated Dipolar Gases
• Nonordinary edge criticaliy of two-dimensional quantum critical magnets
• Topologically-Protected Long Edge Coherence Times in Symmetry-Broken Phases
• RG Flow from ${\varphi }^4$ Theory to the 2D Ising Model
• From one-dimensional charge conserving superconductors to the gapless Haldane phase
• On Analytic Bootstrap for Interface and Boundary CFT
• From trivial to topological paramagnets: The case of ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_2^3$ symmetries in two dimensions
• What is a particle-conserving Topological Superfluid?
• Crystalline Symmetry-Protected Majorana Mode in Number-Conserving Dirac Semimetal Nanowires
• Discrete symmetries of unitary minimal conformal theories
• Spectrum of the Wilson-Fisher conformal field theory on the torus
• Operator product expansion for conformal defects
• Two Point Functions in Defect CFTs
• Interaction induced topological protection in one-dimensional conductors
• Number-conserving interacting fermion models with exact topological superconducting ground states
• Flux-Stabilized Majorana Zero Modes in Coupled One-Dimensional Fermi Wires
• Comments on a state-operator correspondence for the torus
• Exotic criticality in the dimerized spin-1 $XXZ$ chain with single-ion anisotropy
• Topological Edge and Interface states at Bulk disorder-to-order Quantum Critical Points
• Torus Spectroscopy of the Gross-Neveu-Yukawa Quantum Field Theory: Free Dirac versus Chiral Ising Fixed Point
• Ising anyonic topological phase of interacting Fermions in one dimension
• Spectrum of conformal gauge theories on a torus
• Topological phases of parafermionic chains with symmetries
• On the boundary Ising model with disorder operators
• Boundary Criticality of Topological Quantum Phase Transitions in $2d$ systems

Cited by in corpus (102)

• Statistical localization: from strong fragmentation to strong edge modes
• Intrinsically Gapless Topological Phases
• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases
• Topological Transition on the Conformal Manifold
• Topological Holography: Towards a Unification of Landau and Beyond-Landau Physics
• The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap
• Non-Invertible Duality Transformation Between SPT and SSB Phases
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains
• Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality
• Boundary Criticality of the 3D O($N$) Model: From Normal to Extraordinary
• Bulk-boundary correspondence for intrinsically-gapless SPTs from group cohomology
• Subsystem Non-Invertible Symmetry Operators and Defects
• Pivot Hamiltonians as generators of symmetry and entanglement
• Gauging the Kitaev chain
• Topological quantum critical points in the extended Bose-Hubbard model
• Probing sign structure using measurement-induced entanglement
• Spin many-body phases in standard and topological waveguide QED simulators
• Decorated Defect Construction of Gapless-SPT States
• Topological holography, quantum criticality, and boundary states
• Universal entanglement spectrum in gapless symmetry protected topological states
• From trivial to topological paramagnets: The case of ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_2^3$ symmetries in two dimensions
• Intrinsically/Purely Gapless-SPT from Non-Invertible Duality Transformations
• Building models of topological quantum criticality from pivot Hamiltonians
• Duality, Criticality, Anomaly, and Topology in Quantum Spin-1 Chains
• Lattice T-duality from non-invertible symmetries in quantum spin chains
• Signatures of topological phase transition on a quantum critical line
• Quantum phase transition and critical behavior between the gapless topological phases
• Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup
• Supersolid-Superfluid phase separation in the extended Bose-Hubbard model
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Interacting second-order topological insulators in one-dimensional fermions with correlated hopping
• Bulk-boundary correspondence and singularity-filling in long-range free-fermion chains
• Emergent conformal boundaries from finite-entanglement scaling in matrix product states
• Quantum phase transition between topologically distinct quantum critical points
• Recent progress on quantum simulations of non-standard Bose-Hubbard models
• Topological edge modes and phase transition in the critical fermionic chain with long-range interaction
• Symmetry-Enriched Criticality in a Coupled Spin-Ladder
• Disentangling quantum matter with measurements
• Universal finite-size amplitude and anomalous entanglement entropy of $z=2$ quantum Lifshitz criticalities in topological chains
• Quantum phase transition between symmetry enriched topological phases in tensor-network states
• SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement
• Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip
• Boundary deconfined quantum criticality at transitions between symmetry-protected topological chains
• Long-range entanglement and topological excitations
• Topological phase transition between non-high symmetry critical phases and curvature function renormalization group
• Quantization of topological indices in critical chains at low temperatures
• Long-range nonstabilizerness and phases of matter
• Higgs-Confinement Transitions in QCD from Symmetry Protected Topological Phases
• Quantum entanglement of fermionic symmetry-enriched quantum critical points in one dimension
• Boundary SymTFT
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Integrable spin chains and the Clifford group
• Interplay of entanglement structures and stabilizer entropy in spin models
• Gapless Symmetry-Protected Topological States in Measurement-Only Circuits
• Finite Temperature Entanglement Negativity of Fermionic Symmetry Protected Topological Phases and Quantum Critical Points in One Dimension
• Superuniversality from disorder at two-dimensional topological phase transitions
• Gapless higher-order topology and corner states in Floquet systems
• Pivoting through the chiral-clock family
• Classical origins of Landau-incompatible transitions
• Converting electrons into emergent fermions at a superconductor-Kitaev spin liquid interface
• The ordinary surface universality class of the three-dimensional O($N$) model
• Topological metals constructed by sliding quantum wire arrays
• Topological quantum criticality from multiplicative topological phases
• Stabilizer Rényi Entropy Encodes Fusion Rules of Topological Defects and Boundaries
• Topological phases arising from attractive interaction and pair hopping in the Extended Hubbard Model
• Gauge theory and mixed state criticality
• Self-$G$-ality in 1+1 dimensions
• Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains
• Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence
• Interaction-induced phase transitions at topological quantum criticality of an extended Su-Schrieffer-Heeger model
• Universal finite-size scaling in the extraordinary-log boundary phase of three-dimensional $O(N)$ model
• Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant
• Boundary operator expansion and extraordinary phase transition in the tricritical O(N) model
• Boundary and defect criticality in topological insulators and superconductors
• Charge pumps, boundary modes, and the necessity of unnecessary criticality
• The global phase diagram of the cluster-XY spin chain with dissipation
• ${\mathbb{Z}}_2$ gauging and self-dualities of the $XX$ model and its cousins
• Quasiperiodicity protects quantized transport in disordered systems without gaps
• Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions
• Many-body quantum catalysts for transforming between phases of matter
• Universal time evolution of string order parameter in quantum critical systems with boundary invertible or non-invertible symmetry breaking
• Boundary phase transitions of two-dimensional quantum critical XXZ model
• Deconfined criticality as intrinsically gapless topological state in one dimension
• Rigorous lower bound on dynamical exponents in gapless frustration-free systems
• From Hermitian critical to non-Hermitian point-gapped phases
• Boundary operator product expansion coefficients of the three-dimensional Ising universality class
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Generalized Li-Haldane Correspondence in Critical Dirac-Fermion Systems
• Worldline deconfinement and emergent long-range interaction in the entanglement Hamiltonian and in the entanglement spectrum
• Impurity screening by defects in (1+1)$d$ quantum critical systems
• Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle
• Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories
• Emergent gapless spiral phases and conformal Lifshitz criticality in the cluster Ising model with off-diagonal interactions
• Measurement-Induced Entanglement in Conformal Field Theory
• Topology and edge modes surviving criticality in non-Hermitian Floquet systems
• Boundary criticality in two-dimensional interacting topological insulators
• Topology meets symmetry breaking: Hidden order, intrinsically gapless topological states and finite-temperature topological transitions
• Higher-form anomalies and state-operator correspondence beyond conformal invariance
• Topologically nontrivial multicritical points
• Entanglement spectrum of gapless topological phases: a case study with topological superconductors
• Anomalous Dynamical Scaling at Topological Quantum Criticality