Translationally Covariant Modulated Symmetries: Classification and Goldstone

Bo-Ting Chen; Zihan Zhou; Biao Lian

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arXiv · 2606.07952

Translationally Covariant Modulated Symmetries: Classification and Goldstone

Bo-Ting Chen·Zihan Zhou·Biao Lian

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Abstract

Modulated symmetries are global symmetries with a spatially dependent unit of charge, such as the dipole symmetry and the exponential symmetry. We give the generic condition for a modulated symmetry to be compatible with translationally symmetric Hamiltonians, which we define as a translationally covariant modulated symmetry (TCMS). For Abelian TCMSs, we prove that their units of charge can only contain multipole, exponential and harmonic components. Particularly, we classify all the one-dimensional TCMSs by real Jordan normal form blocks. We further derive the generic Goldstone action for SSB phases of continuous TCMSs, by which we show that a broken multipole symmetry gives higher-order gapless Goldstone modes, a broken harmonic symmetry gives gapless Goldstone modes at finite momenta, and a broken exponential symmetry gives no gapless Goldstone modes, modifying the conventional Goldstone theorem.

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Bo-Ting Chen, Zihan Zhou, Biao Lian. 2026-06-06. Translationally Covariant Modulated Symmetries: Classification and Goldstone. https://arxiv.org/abs/2606.07952

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