A 2023 paper by Han, Lake, Lam, Verresen and You introduced one-dimensional symmetry-protected topological (SPT) phases protected by spatially modulated symmetries — dipole, quadrupole, and exponentially modulated — via exactly solvable stabilizer models, and introduced the notion of a bundle symmetry, a symmetry defined patchwise on the chain with transition functions on overlaps, to describe the protecting structure on periodic chains. The paper explicitly flagged open directions including the nature of phase transitions of modulated SPT states into trivial states and the accompanying critical theories, the mappings between modulated SPT and symmetry-breaking phases, realistic spin-model realizations, and the exploration of bundle symmetries in broader circumstances. Subsequent literature has largely resolved two questions in this area: the classification of one-dimensional modulated-symmetry SPT phases, settled in group-cohomology and matrix-product-state terms for dipolar and general multipole symmetries, and the construction of duality mappings between modulated SPT and symmetry-breaking phases via noninvertible Kramers-Wannier and Kennedy-Tasaki transformations, with the gauging of modulated symmetries developed systematically. The surviving open core consists of three linked questions: the generic nature and universality classes of phase transitions out of modulated-SPT order (the one numerically studied case is first-order); the construction of physically realistic, non-stabilizer microscopic realizations of dipolar SPT order together with experimental signatures; and the structural theory of bundle symmetries — whether they can spontaneously break and whether nontrivial higher-dimensional bundle topologies exist.

Background

Symmetry-protected topological (SPT) phases are gapped phases of matter with short-range entanglement that become trivial if the protecting global symmetry is abandoned but otherwise show robust boundary signatures; in one dimension, bosonic SPT phases with an ordinary onsite symmetry group GG are classified by the projective representations of GG acting on the chain edges, equivalently by the second cohomology group H2(G,U(1))H^2(G,U(1)). Canonical examples are the spin-1 Haldane chain (protected by Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2 spin rotations) and the ZN\mathbb{Z}_N cluster state built from commuting stabilizers (local operators that all commute with one another and define the state as their common eigenstate).

A modulated (multipole) symmetry is a global symmetry whose operator carries a position-dependent charge: for a chain of ZN\mathbb{Z}_N clock variables Xj,ZjX_j,Z_j (a discrete NN-state generalization of a qubit, in which XjX_j cyclically shifts the state and ZjZ_j reads out which of the NN values is held), the dipole symmetry is generated by D=∏jXjjD=\prod_j X_j^j alongside the ordinary charge Q=∏jXjQ=\prod_j X_j, so that charge can move only in neutral dipoles. Such symmetries arise in fracton-adjacent systems (systems whose excitations have restricted mobility, with immobile 'fracton' quasiparticles as the extreme case) and are approximately realized in tilted optical lattices (optical-lattice experiments with a potential-energy gradient that suppresses single-particle tunneling, leaving correlated, dipole-preserving hops dominant); their algebra is intertwined with spatial translation, T−1DT=QD\mathcal{T}^{-1}D\mathcal{T}=QD, so they are not onsite symmetries. Han, Lake, Lam, Verresen and You introduced one-dimensional SPT phases protected by such symmetries: exactly solvable stabilizer models (Hamiltonians built from mutually commuting local operators, so that the ground states are known exactly) for the ZN×ZN\mathbb{Z}_N\times\mathbb{Z}_N dipolar, the quadrupolar (∏jXjj2\prod_j X_j^{j^2}), and the exponentially modulated (∏jXjaj\prod_j X_j^{a^j}) cases, each diagnosed by edge-mode degeneracy, string order parameters (nonlocal expectation values joining two chain ends), and symmetry fractionalization (the projective, edge-dependent manner in which the symmetry acts on the entanglement degrees of freedom at the chain ends), and shown to be inequivalent to monopolar SPT phases (SPT phases protected by an ordinary onsite charge symmetry) at the level of matrix product states (a one-dimensional variational wavefunction class built from local tensors).

Two structural features of that work frame the open questions. First, on a periodic chain the dipole operator is globally well defined only when the chain length is divisible by NN; the protecting symmetry is instead a bundle symmetry — symmetry operators defined only on patches of the chain, glued by transition functions (here charge operators) on overlaps, in analogy with gauge fields on fiber bundles. Strictly speaking, the phases are bundle-symmetry protected. Second, when the authors destabilized their N=3N=3 dipolar stabilizer Hamiltonian by a Zeeman term (here a symmetry-preserving external-field coupling of the form ∑j(Xj+Xj†)\sum_j (X_j + X_j^{\dagger}), which commutes with the charge and dipole generators), an exact self-duality (the Hamiltonian maps to itself, up to inversion of the interpolation parameter, under an exact duality transformation) pinned any direct transition to the midpoint of the interpolation, and infinite-system DMRG (a tensor-network ground-state method) found a first-order transition there — a single data point on a question the authors left open in general.

Subsequent work has resolved the mapping direction the paper flagged and completed the classification program initiated by its exactly solvable models. The classification of one-dimensional modulated-symmetry SPT phases is now understood in matrix-product-state and cohomological terms: for a finite abelian group GG the dipolar SPT phases are classified by H2(G×G,U(1))/H2(G,U(1))2H^2(G\times G,U(1))/H^2(G,U(1))^2, with response field theories given by twisted finite tensor gauge theories (lattice gauge theories whose gauge fields couple to extended, higher-moment charge rather than point charges; Lam 2023); this was extended to all rr-pole (multipole) symmetries by Saito, Cao, Han and Ebisu (2025) and to general modulated symmetries, including combinations with spatial symmetries via the crystalline equivalence principle (the principle that a crystalline symmetry acting on a phase can be traded for an internal symmetry acting on an augmented lattice, so that spatially protected phases are classified like ordinary SPT phases), by Anakru, Li, Srinivasan and Bi (2026), Ning, Ebisu and Lam (2026), and Bulmash (2025). Separately, the mapping between modulated SPT phases and symmetry-breaking phases — worked out in the source only for the exponential case — was constructed for the dipolar and exponential cases through noninvertible Kramers-Wannier and Kennedy-Tasaki duality transformations (the Kramers-Wannier duality relating ordered and disordered lattice models, and its nonunitary finite-depth counterpart that maps SPT phases to symmetry-breaking ones; Kim, You and Han 2025; Pace, Delfino, Lam and Aksoy 2025), and the gauging of modulated symmetries has been developed systematically (Kim, Lee and Han 2025; Ebisu and Han 2024).

Problem Statement

The surviving open core of the source's outlook program, after the classification and duality advances recorded above, is to extend the theory of modulated-symmetry SPT chains beyond exactly solvable stabilizer fixed points. It comprises three linked research fronts that together advance these phases as physical phases of matter — with known criticality (1), realizable microscopic origin (2), and an understood protecting symmetry structure (3):

(1) Phase transitions and criticality. Determine the generic nature of quantum phase transitions out of modulated-symmetry SPT phases (dipolar, quadrupolar, exponentially modulated) into trivial paramagnets or symmetry-breaking phases under symmetry-preserving destabilizations: whether such transitions are generically first-order, as in the single N=3N=3 dipolar-plus-Zeeman model studied numerically in the source, or can be continuous; and if continuous, identify the universality classes and critical theories, in particular whether they are distinct from known monopolar SPT transitions such as boundary-deconfined criticality (a critical point at an SPT boundary at which degrees of freedom normally confined to the chain ends deconfine into the bulk). The two alternatives are distinct sub-questions and either may be resolved independently: establishing that a transition can be continuous, together with its universality class, is a separate question from establishing whether such transitions are generically first-order across the dipolar, quadrupolar and exponentially modulated cases.

(2) Physically realistic realizations. Construct microscopic, non-stabilizer models — the analogue of the spin-1 Haldane chain's role for the Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2 SPT — that realize, for example, ZN×ZN\mathbb{Z}_N\times\mathbb{Z}_N dipolar SPT order in a generic (not exactly solvable) Hamiltonian, and characterize the observable consequences, including candidate experimental platforms and measurable signatures of dipolar SPT order as distinct from mere dipole conservation. The named ZN×ZN\mathbb{Z}_N\times\mathbb{Z}_N dipolar case is an illustrative and sufficient target for this front; a construction covering all modulated symmetries is not required.

(3) Bundle-symmetry structure. Develop the structural theory of bundle symmetries themselves: whether and how a bundle symmetry can spontaneously break, and whether nontrivial bundle-symmetry topologies — for example higher-dimensional analogues with monopole-like configurations — exist and constrain the phases they protect. (The gauging of modulated symmetries is largely understood from the works cited above; the spontaneous-breaking and higher-dimensional-topology questions remain open.) The spontaneous-breaking question and the higher-dimensional-topology question are distinct sub-questions and either may be resolved independently.

These three fronts share one objective — completing the physical theory of modulated-symmetry SPT order initiated by the stabilizer fixed-point models — and they are methodologically coupled: the same symmetry algebra (charge, multipole, translation) and the same tensor-network diagnostics govern all three. Each front is separately valuable and independently refutable: resolving any one of them leaves the other two open, and together they form the surviving open program flagged by the source.

References

  1. Jung Hoon Han, Ethan Lake, Ho Tat Lam, Ruben Verresen, Yizhi You, 'Topological quantum chains protected by dipolar and other modulated symmetries', Phys. Rev. B 109, 125121 (2024), DOI 10.1103/PhysRevB.109.125121, arXiv:2309.10036, https://arxiv.org/abs/2309.10036
  2. Ho Tat Lam, 'Classification of dipolar symmetry-protected topological phases: Matrix product states, stabilizer Hamiltonians and finite tensor gauge theories', Phys. Rev. B 109, 115142 (2024), DOI 10.1103/PhysRevB.109.115142, arXiv:2311.04962, https://arxiv.org/abs/2311.04962
  3. Takuma Saito, Weiguang Cao, Bo Han, Hiromi Ebisu, 'Matrix product state classification of one-dimensional multipole symmetry-protected topological phases', Phys. Rev. B 112, 195133 (2025), DOI 10.1103/6txg-7hy7, arXiv:2509.09244, https://arxiv.org/abs/2509.09244
  4. Amogh Anakru, Linhao Li, Sarvesh Srinivasan, Zhen Bi, 'Matrix Product States for Modulated Symmetries: SPT, LSM, and Beyond', arXiv:2603.19189, https://arxiv.org/abs/2603.19189
  5. Shang-Qiang Ning, Hiromi Ebisu, Ho Tat Lam, 'Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints', arXiv:2603.19381, https://arxiv.org/abs/2603.19381
  6. Daniel Bulmash, 'Defect Networks for Topological Phases Protected By Modulated Symmetries', arXiv:2508.06604, https://arxiv.org/abs/2508.06604
  7. Jintae Kim, Yizhi You, Jung Hoon Han, 'Noninvertible symmetry and topological holography for modulated SPT in one dimension', SciPost Phys. 19, 110 (2025), DOI 10.21468/SciPostPhys.19.4.110, arXiv:2507.02324, https://arxiv.org/abs/2507.02324
  8. Salvatore D. Pace, Guilherme Delfino, Ho Tat Lam, Ömer Mert Aksoy, 'Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections', SciPost Phys. 18, 021 (2025), DOI 10.21468/SciPostPhys.18.1.021, arXiv:2406.12962, https://arxiv.org/abs/2406.12962
  9. Jintae Kim, Jong Yeon Lee, Jung Hoon Han, 'Gauging modulated symmetries via multiple gauge symmetry operators and adaptive quantum circuits', Phys. Rev. B 113, 085139 (2026), DOI 10.1103/xg4h-vqv1, arXiv:2503.15834, https://arxiv.org/abs/2503.15834
  10. Hiromi Ebisu, Bo Han, 'Noninvertible operators in one, two, and three dimensions via gauging spatially modulated symmetry', Phys. Rev. B 111, 035149 (2025), DOI 10.1103/PhysRevB.111.035149, arXiv:2409.16744, https://arxiv.org/abs/2409.16744
  11. Donghae Seo, Gil Young Cho, Robert-Jan Slager, 'Noninvertible duality and symmetry topological order of one-dimensional lattice models with spatially modulated symmetry', Phys. Rev. B 111, 115112 (2025), DOI 10.1103/PhysRevB.111.115112, arXiv:2411.04182, https://arxiv.org/abs/2411.04182
  12. Bo-Ting Chen, Zihan Zhou, Biao Lian, 'Translationally Covariant Modulated Symmetries: Classification and Goldstone', arXiv:2606.07952, https://arxiv.org/abs/2606.07952
  13. Hiromi Ebisu, Bo Han, Weiguang Cao, 'Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies', SciPost Phys. 20, 117 (2026), DOI 10.21468/SciPostPhys.20.4.117, arXiv:2510.18689, https://arxiv.org/abs/2510.18689
  14. Amogh Anakru, Zhen Bi, 'Topological Phases and Phase Transitions with Dipolar Symmetry Breaking', arXiv:2403.19601, https://arxiv.org/abs/2403.19601
  15. Seungho Lee, Daesik Kim, Hyun-Yong Lee, Jung Hoon Han, 'Projector, Neural, and Tensor-Network Representations of ZN\mathbb{Z}_N Cluster and Dipolar-cluster SPT States', arXiv:2604.06741, https://arxiv.org/abs/2604.06741
  16. Nick G. Jones, Ryan Thorngren, Ruben Verresen, Abhishodh Prakash, 'Charge pumps, pivot Hamiltonians and symmetry-protected topological phases', Phys. Rev. B 112, 165123 (2025), DOI 10.1103/rtq1-pplf, arXiv:2507.00995, https://arxiv.org/abs/2507.00995
  17. Yizhi You, Masaki Oshikawa, 'Intrinsic symmetry-protected topological mixed state from modulated symmetries and hierarchical structure of boundary anomaly', Phys. Rev. B 110, 165160 (2024), DOI 10.1103/PhysRevB.110.165160, arXiv:2407.08786, https://arxiv.org/abs/2407.08786
  18. Saranesh Prembabu, Ryan Thorngren, Ruben Verresen, 'Boundary-deconfined quantum criticality at transitions between symmetry-protected topological chains', Phys. Rev. B 109, L201112 (2024), DOI 10.1103/PhysRevB.109.L201112, arXiv:2208.12258, https://arxiv.org/abs/2208.12258