Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases

Abstract

We propose and prove a family of generalized Lieb-Schultz-Mattis (LSM) theorems for symmetry protected topological (SPT) phases on boson/spin models in any dimensions. The "conventional" LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a nontrivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or "higher-order" crystalline SPT phases, characterized by nontrivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.

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• Lieb-Schultz-Mattis theorems for symmetry protected topological phases
• Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries
• (SPT-)LSM theorems from projective non-invertible symmetries
• Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Definition and Classification of Fermi Surface Anomalies
• Average-exact mixed anomalies and compatible phases
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• Filling constraints on translation invariant dipole conserving systems
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• Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
• Entanglement-enabled symmetry-breaking orders
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly