Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases

Abstract

The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional generalizations by Oshikawa and Hastings establish that a translation-invariant lattice model of spin-$1/2$'s can not have a non-degenerate ground state preserving both spin and translation symmetries. Recently it was shown that LSM theorems can be interpreted in terms of bulk-boundary correspondence of certain weak symmetry-protected topological (SPT) phases. In this work we discuss LSM-type theorems for two-dimensional fermionic systems, which have no bosonic analogs. They follow from a general classification of weak SPT phases of fermions in three dimensions. We further derive constraints on possible gapped symmetry-enriched topological phases in such systems. In particular, we show that lattice translations must permute anyons, thus leading to "symmetry-enforced" non-Abelian dislocations, or "genons". We also discuss surface states of other weak SPT phases of fermions.

arXiv commentv2: typo fixed, acknowledgement updated; v3 published version

References in corpus (50)

• Anyons in an exactly solved model and beyond
• Quantum Spin Liquids
• Quantum Spin Liquid States
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Topological phases of fermions in one dimension
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Classifying quantum phases using Matrix Product States and PEPS
• Lieb-Schultz-Mattis in Higher Dimensions
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Commensurability, excitation gap and topology in quantum many-particle systems on a periodic lattice
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Topological approach to Luttinger's theorem and the Fermi surface of a Kondo lattice
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Genons, twist defects, and projective non-Abelian braiding statistics
• Topological Nematic States and Non-Abelian Lattice Dislocations
• Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface
• Filling constraints for spin-orbit coupled insulators in symmorphic and non-symmorphic crystals
• Fermionic SPT phases in higher dimensions and bosonization
• Topological Order and Absence of Band Insulators at Integer Filling in Non-Symmorphic Crystals
• Building crystalline topological phases from lower-dimensional states
• Gauging spatial symmetries and the classification of topological crystalline phases
• Anomalous Symmetry Fractionalization and Surface Topological Order
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
• Fermionic Modular Categories and the 16-fold Way
• Symmetry fractionalization and twist defects
• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• Fermionic Matrix Product States and One-Dimensional Topological Phases
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Lieb-Schultz-Mattis Theorem and its generalizations from the Perspective of the Symmetry Protected Topological phase
• Towards a complete classification of fermionic symmetry protected topological phases in 3D and a general group supercohomology theory
• All Majorana Models with Translation Symmetry are Supersymmetric
• Constraints on topological order in Mott Insulators
• Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions
• Minimalist approach to the classification of symmetry protected topological phases
• Unconventional Fusion and Braiding of Topological Defects in a Lattice Model
• Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions
• Weak symmetry breaking in two dimensional topological insulators
• Dyonic Lieb-Shultz-Mattis Theorem and Symmetry Protected Topological Phases in Decorated Dimer Models
• Spin Topological Field Theory and Fermionic Matrix Product States
• Lieb-Schultz-Mattis theorems for symmetry protected topological phases
• Flux-fusion anomaly test and bosonic topological crystalline insulators
• Fermionic Matrix Product States and One-Dimensional Short-Range Entangled Phases with Anti-Unitary Symmetries
• Classification of symmetry fractionalization in gapped ${\mathbb{Z}}_2$ spin liquids
• Surface Topological Order and a new 't Hooft Anomaly of Interaction Enabled 3+1D Fermion SPTs
• Anomalous Quasiparticle Symmetries and Non-Abelian Defects on Symmetrically Gapped Surfaces of Weak Topological Insulators

Cited by in corpus (49)

• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Emergent Symmetry and Conserved Current at a One Dimensional Incarnation of Deconfined Quantum Critical Point
• Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
• Anomaly matching and symmetry-protected critical phases in $SU(N)$ spin systems in 1+1 dimensions
• Towards a classification of mixed-state topological orders in two dimensions
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Fermionic symmetry fractionalization in (2+1)D
• Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems
• Non-zero momentum requires long-range entanglement
• Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
• Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Quantum many-body topology of quasicrystals
• On gapped boundaries for SPT phases beyond group cohomology
• Rotation Symmetry-Protected Topological Phases of Fermions
• Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• Lieb-Schultz-Mattis type constraints on Fractonic Matter
• Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions
• AKLT models on decorated square lattices are gapped
• Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications
• A UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases
• Intrinsically Interacting Higher-Order Topological Superconductors
• Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
• (SPT-)LSM theorems from projective non-invertible symmetries
• Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Strongly correlated crystalline higher-order topological phases in two-dimensional systems: A coupled-wire study
• Low energy effective field theories of fermion liquids and mixed $U(1)\times {\mathbb{R}}^d$ anomaly
• Definition and Classification of Fermi Surface Anomalies
• Translation symmetry-enriched toric code insulator
• Bootstrapping Lieb-Schultz-Mattis anomalies
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension
• Interacting Edge States of Fermionic Symmetry-Protected Topological Phases in Two Dimensions
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Filling constraints on translation invariant dipole conserving systems
• Exactly solvable lattice models for interacting electronic insulators in two dimensions
• Mirror anomaly in fermionic topological orders
• Anomaly constraint on chiral central charge of (2+1)d topological order
• Universal contributions to charge fluctuations in spin chains at finite temperature
• Numerical identification and gapped boundaries of abelian fermionic topological order
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Ordering the topological order in the fractional quantum Hall effect
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly