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arXiv:2606.07952

Translationally Covariant Modulated Symmetries: Classification and Goldstone

Bo-Ting Chen, Zihan Zhou, Biao Lian

Published Jun 6, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2606.07952

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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.
Spatially modulated symmetries are one of the new types of symmetries the symmetry actions of which are position dependent. Yet exotic phases resulting from these spatially modulated symmetries are not fully understood and classified. In this work, we systematically classify one-dimensional bosonic symmetry-protected topological phases of multipole symmetries by employing the matrix product state formalism. The symmetry action induces projective representations at the ends of an open chain, which we identify via group cohomology. In particular, for rrr-pole symmetries---for instance, r=0r=0r=0 (global), 1 (dipole), and 2 (quadrupole)---the classification is determined by distinct components of second cohomology groups that encode the boundary projective representations.
We consider systems in which a continuous symmetry GGG, which may be anomalous, is spontaneously broken to an anomaly-free subgroup HHH such that the effective action for the Goldstone modes contains topologically nontrivial terms. If the original system has trivial GGG anomaly, then it is known that the possible topological terms are fully determined by SPT or SET invariants of the residual HHH symmetry. Here we address the more general setting in which the GGG symmetry has an anomaly. We argue that in general, the appropriate concept to consider is the ``compatibility relation'' between the Goldstone invariants and the GGG anomaly. In the case where the Goldstone modes can be gapped out to obtain invertible families (i.e., without any topological order), we give an explicit mathematical scheme to construct the desired compatibility relation. We also address the case where gapping out the Goldstone modes leads to a family of topologically ordered states. We discuss several examples including the canonical Thouless pump, the quantum Hall ferromagnet, pumps arising from breaking U(1)\mathrm{U}(1)U(1) symmetry at the boundary of topological insulators in two and three dimensions, and pumps classified by the higher Chern number.
We investigate the physics of one-dimensional symmetry-protected topological (SPT) phases protected by symmetries whose symmetry generators exhibit spatial modulation. We focus in particular on phases protected by symmetries with linear (i.e., dipolar), quadratic, and exponential modulations. We present a simple recipe for constructing modulated SPT models by generalizing the concept of decorated domain walls to spatially modulated symmetry defects, and develop several tools for characterizing and classifying modulated SPT phases. A salient feature of modulated symmetries is that they are generically only present for open chains, and are broken upon the imposition of periodic boundary conditions. Nevertheless, we show that SPT order is present even with periodic boundary conditions, a phenomenon we understand within the context of an object we dub a ``bundle symmetry.'' In addition, we show that modulated SPT phases can avoid a certain no-go theorem, leading to an unusual algebraic structure in their matrix product state descriptions.
Fractonic phases are new phases of matter that host excitations with restricted mobility. We show that a certain class of gapless fractonic phases are realized as a result of spontaneous breaking of continuous higher-form symmetries whose conserved charges do not commute with spatial translations. We refer to such symmetries as nonuniform higher-form symmetries. These symmetries fall within the standard definition of higher-form symmetries in quantum field theory, and the corresponding symmetry generators are topological. Worldlines of particles are regarded as the charged objects of 1-form symmetries, and mobility restrictions can be implemented by introducing additional 1-form symmetries whose generators do not commute with spatial translations. These features are realized by effective field theories associated with spontaneously broken nonuniform 1-form symmetries. At low energies, the theories reduce to known higher-rank gauge theories such as scalar/vector charge gauge theories, and the gapless excitations in these theories are interpreted as Nambu-Goldstone modes for higher-form symmetries. Due to the nonuniformity of the symmetry, some of the modes acquire a gap, which is the higher-form analogue of the inverse Higgs mechanism of spacetime symmetries. The gauge theories have emergent nonuniform magnetic symmetries, and some of the magnetic monopoles become fractonic. We identify the ’t Hooft anomalies of the nonuniform higher-form symmetries and the corresponding bulk symmetry-protected topological phases. By this method, the mobility restrictions are fully determined by the choice of the commutation relations of charges with translations. This approach allows us to view existing (gapless) fracton models such as the scalar/vector charge gauge theories and their variants from a unified perspective and enables us to engineer theories with desired mobility restrictions.
We investigate the physics of one-dimensional symmetry protected topological (SPT) phases protected by symmetries whose symmetry generators exhibit spatial modulation. We focus in particular on phases protected by symmetries with linear (i.e., dipolar), quadratic and exponential modulations. We present a simple recipe for constructing modulated SPT models by generalizing the concept of decorated domain walls to spatially modulated symmetry defects, and develop several tools for characterizing and classifying modulated SPT phases. A salient feature of modulated symmetries is that they are generically only present for open chains, and are broken upon the imposition of periodic boundary conditions. Nevertheless, we show that SPT order is present even with periodic boundary conditions, a phenomenon we understand within the context of an object we dub a ``bundle symmetry''. In addition, we show that modulated SPT phases can avoid a certain no-go theorem, leading to an unusual algebraic structure in their matrix product state descriptions.
When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu-Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy, particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be fewer than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulas for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. In the process we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb-Schultz-Mattis theorem.