Lieb-Schultz-Mattis Theorem and its generalizations from the Perspective of the Symmetry Protected Topological phase

Abstract

We ask whether a local Hamiltonian with a featureless (fully gapped and nondegenerate) ground state could exist in certain quantum spin systems. We address this question by mapping the vicinity of certain quantum critical point (or gapless phase) of the $d-$dimensional spin system under study to the boundary of a $(d+1)-$dimensional bulk state, and the lattice symmetry of the spin system acts as an on-site symmetry in the field theory that describes both the selected critical point of the spin system, and the corresponding boundary state of the $(d+1)-$dimensional bulk. If the symmetry action of the field theory is nonanomalous, i.e. the corresponding bulk state is a trivial state instead of a bosonic symmetry protected topological (SPT) state, then a featureless ground state of the spin system is allowed; if the corresponding bulk state is indeed a nontrivial SPT state, then it likely excludes the existence of a featureless ground state of the spin system. From this perspective we identify the spin systems with SU($N$) and SO($N$) symmetries on one, two and three dimensional lattices that permit a featureless ground state. We also verify our conclusions by other methods, including an explicit construction of these featureless spin states.

15 pages, 5 figures

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• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
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• Symmetric Fracton Matter: Twisted and Enriched
• Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
• Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
• Obstructions to Gapped Phases from Non-Invertible Symmetries
• Fermionic Lieb-Schultz-Mattis Theorems and Weak Symmetry-Protected Phases
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• Emergent Symmetry and Conserved Current at a One Dimensional Incarnation of Deconfined Quantum Critical Point
• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
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• Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models
• Anomaly matching and symmetry-protected critical phases in $SU(N)$ spin systems in 1+1 dimensions
• C-P-T anomaly matching in bosonic quantum field theory and spin chains
• Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries
• Deconfined Quantum Critical Point on the Triangular Lattice
• Non-zero momentum requires long-range entanglement
• Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries
• Rotation Symmetry-Protected Topological Phases of Fermions
• Unquantized anomalies in topological semimetals
• Emergent anomalies and generalized Luttinger theorems in metals and semimetals
• Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains
• Lieb-Schultz-Mattis type constraints on Fractonic Matter
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Physics of Symmetry Protected Topological phases involving Higher Symmetries and their Applications
• AKLT models on decorated square lattices are gapped
• A UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases
• Fermi Surface Symmetric Mass Generation
• Bulk and edge dynamics of a 2D Affleck-Kennedy-Lieb-Tasaki model
• (SPT-)LSM theorems from projective non-invertible symmetries
• Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories
• Filling constraints on fermionic topological order in zero magnetic field
• Generic "Unnecessary" Quantum Critical Points with Minimal Degrees of Freedom
• Average-exact mixed anomalies and compatible phases
• Boundary conditions and anomalies of conformal field theories in 1+1 dimensions
• Global anomaly matching in higher-dimensional ${\mathbb{CP}}^{N-1}$ model
• Filling constraints on translation invariant dipole conserving systems
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Translation constraints on quantum phases with twisted boundary conditions
• Theory of Dirac Spin-Orbital Liquids: monopoles, anomalies, and applications to $SU(4)$ honeycomb models
• SU(4) topological RVB spin liquid on the square lattice
• Classical origins of Landau-incompatible transitions
• Lattice regularizations of $\theta$ vacua: Anomalies and qubit models
• Anomalous domain wall condensation in a modified Ising chain
• (3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases
• Anomaly constraint on chiral central charge of (2+1)d topological order
• Universal contributions to charge fluctuations in spin chains at finite temperature
• Building Symmetry Enriched Topological Phases from a Bipartite Lattice Construction and Anyon Condensation
• Lieb-Schultz-Mattis constraints for the insulating phases of the one-dimensional SU($N$) Kondo lattice model
• Absence of localization in Weyl semimetals
• Interacting Crystalline Topological Insulators in two-dimensions with Time-Reversal Symmetry
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Anomalies of global symmetries on the lattice
• Anomaly-induced edge currents in hydrodynamics with parity anomaly
• Parity anomaly from LSM: exact valley symmetries on the lattice
• When Wannier centers jump: Critical points between atomic insulating phases