The paper classifies one-dimensional Abelian translationally covariant modulated symmetries via Jordan normal forms and derives their Goldstone actions, which modify the conventional theorem by type of symmetry.
Lian, Quantum breakdown model: From many-body localization to chaos with scars, Physical Review B107, 10.1103/physrevb.107.115171 (2023)
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Translationally Covariant Modulated Symmetries: Classification and Goldstone
The paper classifies one-dimensional Abelian translationally covariant modulated symmetries via Jordan normal forms and derives their Goldstone actions, which modify the conventional theorem by type of symmetry.