Lieb-Schultz-Mattis in Higher Dimensions

Abstract

A generalization of the Lieb-Schultz-Mattis theorem to higher dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range order, in which case there are gapless modes, or have only short-range correlations, in which case there are topological excitations. The result uses a set of loop operators, analogous to those used in gauge theories, defined in terms of the spin operators of the theory. We also obtain various cluster bounds on expectation values for gapped systems. These bounds are used, under the assumption of a gap, to rule out the first case of long-range order, after which we show the existence of a topological excitation. Compared to the ground state, the topologically excited state has, up to a small error, the same expectation values for all operators acting within any local region, but it has a different momentum.

14 pages, 3 figures, final version in press

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• Self-$G$-ality in 1+1 dimensions
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition
• Topological states in strongly correlated systems
• Lieb-Robinson Bounds for the Toda Lattice
• Entanglement subvolume law for 2D frustration-free spin systems
• Dispersive Estimates for Harmonic Oscillator Systems
• Locality of gapped ground states in systems with power-law decaying interactions
• Area law for steady states of detailed-balance local Lindbladians
• Non-invertible symmetries in finite-group gauge theory
• Classical and quantum spin liquids
• Localizable quantum coherence
• Time-Polynomial Lieb-Robinson bounds for finite-range spin-network models
• Simulating spin systems with Majorana networks
• Topological 'Luttinger' invariants protected by non-symmorphic symmetry in semimetals
• Finite-Entanglement Scaling of 2D Metals
• The Chiral Gross-Neveu model on the lattice via a Landau-forbidden phase transition
• No sliding in time
• A Theorem on Extensive Spectral Degeneracy for Systems with Higher Symmetries in General Dimensions
• Strictly local tensor networks for short-range topological insulators
• Nonsymmorphic spin-space cubic groups and SU(2)${}_1$ conformal invariance in one-dimensional spin-1/2 models
• Theory of magnetism for rare-earth magnets on the Shastry-Sutherland lattice with non-Kramers ions
• Projectification of point group symmetries with a background flux and Lieb-Schultz-Mattis theorem
• Diagnosing electronic phases of matter using photonic correlation functions
• Sudden switch of generalized Lieb-Robinson velocity in a transverse field Ising spin chain
• Topological phases of non-symmorphic crystals : Shastry-Sutherland lattice at integer filling
• Evidence for a gapless Dirac spin-liquid ground state in a spin-3/2 triangular-lattice antiferromagnet
• Topological Current in Fractional Chern Insulators
• Phases of frustrated quantum antiferromagnets on the square and triangular lattices
• Logarithmic light cone, slow entanglement growth, and quantum memory
• Ordering the topological order in the fractional quantum Hall effect
• Pathways from a chiral superconductor to a composite Fermi liquid
• Absence of localization in Weyl semimetals
• Ground-State Degeneracy of Correlated Insulators with Edges
• Numerical identification and gapped boundaries of abelian fermionic topological order
• Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
• Generalized Statistics on Lattices
• Bounded light cone and robust topological order out of equilibrium
• Macroscopic Particle Transport in Dissipative Long-Range Bosonic Systems
• Frustrated spin-$\frac{1}{2}$ Heisenberg magnet on an $AA$-stacked honeycomb bilayer: High-order study of the collinear magnetic phases of the $J_1$--$J_2$--$J_1^{\perp }$ model
• Disentangling anomaly-free symmetries of quantum spin chains
• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• Sub-ballistic operator growth in spin chains with heavy-tailed random fields
• A primer on Kitaev Model: Basic aspects, material realization, and recent experiments
• Computing local properties in the trivial phase
• Modulating Hamiltonian Approach to Quantum Many-Body Systems and Crystalline Topological Phases Protected by Generalized Magnetic Translations
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Identification of gapless phases by a twisting operator
• Possible phases of the spin-$1/2$ $XXZ$ model on a honeycomb lattice by boson-vortex duality
• Quantum Chaos in Non-Markovian Open Quantum Systems: Interferometric OTOC, Loschmidt Echo and Commutator Operator Norm
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• Lieb-Robinson bounds with exponential-in-volume tails
• Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Higher Hall conductivity from a single wave function: Obstructions to symmetry-preserving gapped edge of (2+1)D topological order
• Hardware-efficient quantum annealing with error mitigation via classical shadow
• Worldline deconfinement and emergent long-range interaction in the entanglement Hamiltonian and in the entanglement spectrum
• Topological Phase Transitions of Interacting Fermions in the Presence of a Commensurate Magnetic Flux
• Unnecessary quantum criticality in $SU(3)$ kagome magnets
• Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle
• Phase Estimation with Compressed Controlled Time Evolution
• Les Houches Lecture Notes on Tensor Networks
• Correlation-driven non-trivial phases in single bi-layer Kagome intermetallics
• Haldane's Instanton in 2D Heisenberg model revisited: along the Avenue of Topology
• Topologically ordered states in infinite quantum spin systems
• Energy gap of neutral excitations implies vanishing charge susceptibility
• Entanglement-enabled symmetry-breaking orders
• Role of generic scale invariance in a Mott transition from a U(1) spin-liquid insulator to a Landau Fermi-liquid metal
• Ballistic propagation of a local impact in the one-dimensional $XY$ model
• Rate of cluster decomposition via Fermat-Steiner point
• Lieb-Robinson bounds imply locality of interactions
• Bosonic SET to SPT Transition under Dyonic LSM Theorem
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
• Möbius molecules and fragile Mott insulators
• Itinerant magnetism in the triangular lattice Hubbard model at half-doping: Application to twisted transition-metal dichalcogenides
• Anisotropy-induced spin parity effects
• Vestigial Nematic Order at Zero Temperature in Two-Dimensional Frustrated Quantum Antiferromagnets
• Non-invertible translation from Lieb-Schultz-Mattis anomaly
• When Wannier centers jump: Critical points between atomic insulating phases
• Existence of the thermodynamic limit and asymptotic behavior of some irreversible quantum dynamical systems
• On ground state phases of quantum spin systems
• Polarization-based indices in quantum many-body systems: validity and extension beyond one dimension
• Spectral statistics and energy-gap scaling in $k-$local spin Hamiltonians
• Crystallographic Interacting Topological Phases and Equivariant Cohomology: To assume or not to assume
• Quantum Spin Systems after DLS1978
• Excitation and Entanglement Transfer Near Quantum Critical Points
• Symmetry-enforced minimal entanglement and correlation in quantum spin chains
• Knabe's spectral gap method for open boundary conditions
• Automorphic equivalence within gapped phases in the bulk
• Entanglement Complexity in Many-body Systems from Positivity Scaling Laws
• Energy of fermionic ground states with low-entanglement single-reference expansions, and tensor-based strong-coupling extensions of the coupled-cluster method
• Local symmetries and extensive ground-state degeneracy of a 1D supersymmetric fermionic chain
• Itinerant magnetism in Hubbard models with long-range interactions
• Collective effects in an incompressible electronic liquid
• A Personal History of the Hastings-Michalakis Proof of Hall Conductance Quantization
• Anomalies of kagome antiferromagnets on magnetization plateaus