Filling-enforced constraint on the quantized Hall conductivity on a periodic lattice
arXiv:1705.09298 · doi:10.1016/j.aop.2019.168060
Abstract
We discuss quantum Hall effects in a gapped insulator on a periodic two-dimensional lattice. We derive a universal relation among the the quantized Hall conductivity, and charge and flux densities per physical unit cell. This follows from the magnetic translation symmetry and the large gauge invariance, and holds for a very general class of interacting many-body systems. It can be understood as a combination of Laughlin's gauge invariance argument and Lieb-Schultz-Mattis-type theorem. A variety of complementary arguments, based on a cut-and-glue procedure, the many-body electric polarization, and a fractionalization algebra of magnetic translation symmetry, are given. Our universal relation is applied to several examples to show nontrivial constraints. In particular, a gapped ground state at a fractional charge filling per physical unit cell must have either a nonvanishing Hall conductivity or anyon excitations, excluding a trivial Mott insulator.
17+5 pages, 4 figures, published version
References in corpus (12)
- High temperature fractional quantum Hall states
- Fractional quantum Hall states at zero magnetic field
- Nearly-flat bands with nontrivial topology
- Scalar spin chirality and quantum Hall effect on a triangular lattice
- Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
- Symmetry fractionalization and twist defects
- Fractionalization, topological order, and quasiparticle statistics
- Bosonic Integer Quantum Hall effect in an interacting lattice model
- Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
- Filling-Enforced Quantum Band Insulators in Spin-Orbit Coupled Crystals
- Universal Relation among Many-Body Chern Number, Rotation Symmetry, and Filling
- Kane-Mele-Hubbard model on the -flux honeycomb lattice
Cited by in corpus (45)
- Zero-field composite Fermi liquid in twisted semiconductor bilayers
- Non-Fermi liquids as ersatz Fermi liquids: general constraints on compressible metals
- Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
- Electric polarization as a nonquantized topological response and boundary Luttinger theorem
- Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena
- A many-body index for quantum charge transport
- Real-space recipes for general topological crystalline states
- Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
- A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers
- Non-zero momentum requires long-range entanglement
- Higher-form Gauge Symmetries in Multipole Topological Phases
- Classification of Topological Phase Transitions and van Hove Singularity Steering Mechanism in Graphene Superlattices
- Lieb-Schultz-Mattis type constraints on Fractonic Matter
- Doping a fractional quantum anomalous Hall insulator
- Fermi Surface Symmetric Mass Generation
- Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases
- (SPT-)LSM theorems from projective non-invertible symmetries
- -Symmetric Quantum Spin-Orbital Liquids on Various Lattices
- Rational indices for quantum ground state sectors
- Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups
- Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
- Low energy effective field theories of fermion liquids and mixed anomaly
- Definition and Classification of Fermi Surface Anomalies
- Unconventional quantum phase transitions in a one-dimensional Lieb-Schultz-Mattis system
- Rigorous Index Theory for One-Dimensional Interacting Topological Insulators
- Filling constraints on translation invariant dipole conserving systems
- Exactness of linear response in the quantum Hall effect
- A many-body Fredholm index for ground state spaces and Abelian anyons
- Lieb-Schultz-Mattis theorem in higher dimensions from approximate magnetic translation symmetry
- Anomalous domain wall condensation in a modified Ising chain
- Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking
- Projectification of point group symmetries with a background flux and Lieb-Schultz-Mattis theorem
- Apparent inconsistency between Streda formula and Hall conductivity in reentrant integer quantum anomalous Hall effect in twisted MoTe
- Chiral Spin Liquid and Quantum Phase Transition in the Triangular Lattice Hofstadter-Hubbard Model
- Crystalline invariants of fractional Chern insulators
- Lieb-Schultz-Mattis Theorem and the Filling Constraint
- Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
- Ordering the topological order in the fractional quantum Hall effect
- Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
- Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
- Mapping Chern numbers in quasi-periodic interacting spin chains
- Topological Phase Transitions of Interacting Fermions in the Presence of a Commensurate Magnetic Flux
- Theory of Hofstadter Superconductors
- Topological Invariants of a Filling-Enforced Quantum Band Insulator
- Anisotropy-induced spin parity effects