Spatially Modulated Symmetries

• Spatially modulated symmetries are spatially varying internal transformations defined by functions of position (e.g., polynomial or exponential), crucial for enforcing new conservation laws.
• They impose finite-momentum and higher-moment (e.g., dipole) conservation, revealing rich dynamics in density waves, fracton phases, and symmetry-protected topological orders.
• Their unique modulation leads to phenomena like Hilbert space fragmentation, non-invertible dualities, and UV/IR mixing, impacting models from condensed matter to quantum field theory.

Spatially modulated symmetries are symmetry transformations whose action varies in space, often through weights or phase factors that depend nontrivially on the spatial coordinates. Unlike conventional global internal symmetries—which act identically at every point—spatially modulated symmetries can enforce conservation laws at finite momentum, impose higher-moment (such as dipole or multipole) conservation, or act non-uniformly according to prescribed modulation functions (e.g., polynomial, exponential, or quasi-periodic). This framework unifies a broad array of phenomena across condensed matter physics, high energy theory, statistical mechanics, and quantum information, including the physics of density waves, nematic order, fracton phases, and symmetry-protected topological (SPT) phases with unconventional protecting groups.

1. Foundational Notions: Definition, Formal Structure, and Examples

A spatially modulated symmetry is an internal symmetry transformation for which the generator is given by a spatially varying function. The canonical form for a $d$-dimensional lattice is: $U_f = \prod_{j} (U_j)^{f(j)}$ with $U_j$ a symmetry operator (e.g., an on-site clock or spin rotation) and $f(j)$ a function of spatial position. The function $f$ can be polynomial (e.g., $f(j) = j$ for dipole symmetry), exponential ($f(j) = a^j$), periodic ($f(j) = \cos(k^* j + \phi)$), or more generally determined by a recurrence relation (Sala et al., 2021, Pace et al., 18 Jun 2024).

Characteristic properties:

• For a discrete model, $f(j)$ may be taken as an integer-valued function modulo $N$ (for clock models) or a real/complex-valued function for continuous U(1) symmetry.
• Modulated symmetries generalize onsite symmetries and multipole symmetries (Sala et al., 2021, Sala et al., 2023, Han et al., 2023).
• The modulation may lead to conservation of not just total charge but e.g., $\mathcal{Q}_{k^*} = \sum_j e^{i k^* j} n_j$ (finite-momentum density).

Prominent examples include:

• Dipole symmetry: $U_\text{dipole} = \prod_j (U_j)^j$
• Exponential symmetry: $U = \prod_j (U_j)^{a^j}$ (Pace et al., 18 Jun 2024)
• Modulated charge: In bosonic or fermionic chains, operators conserve e.g., $\sum_j \cos(k^* j) n_j$
• Subsystem symmetries: Actions restricted to lower-dimensional subspaces (e.g., row or column flips in 2D) which can be viewed as a spatially modulated symmetry

Physical realizations span:

• Charge density waves and nematic phases with finite-q order
• Models with multipolar conservation (fracton systems, Hilbert space fragmentation)
• Topological phases protected by multipolar or periodically modulated symmetry defects (Han et al., 2023, Bulmash, 8 Aug 2025)

2. Dynamical Consequences and Modulated Symmetry-Enforced Phenomena

The presence of spatially modulated symmetries yields distinct dynamical and static consequences, depending on microscopic implementation and modulation structure:

A. Spontaneous Symmetry Breaking and Goldstone Modes:

In higher-derivative Lorentz-invariant theories, spatial modulation of derivative order parameters leads to unconventional Nambu-Goldstone (NG) modes (Nitta et al., 2017, Nitta et al., 2017). The NG boson associated with the broken symmetry may exhibit a vanishing quadratic kinetic term and arise only through quartic-derivative (“unconventional Goldstone”) dynamics along the modulated direction, with residual gapless Higgs-like excitations orthogonal to the flat direction.

B. Topological Phases and SPT Order:

SPT phases may be protected not only by uniform internal symmetries but by modulated symmetry operations. For example, in 1D, the dipolar symmetry ($\prod_j X_j^j$) protects “multipolar” SPT order with unique edge mode algebra and “bundle symmetries” to resolve global definition issues on closed geometries (Han et al., 2023). The construction and classification of such modulated-Symmetry Protected Topological (MSPT) phases require substantial modifications to the crystalline equivalence principle and defect network formalism (Bulmash, 8 Aug 2025).

C. Hilbert Space Fragmentation and Dynamics:

In rotor/bosonic chains, conservation of modulated charges at incommensurate momentum results in Hilbert space fragmentation into exponentially many disconnected sectors (“quantum many-body scar”–like structure), even with infinite on-site Hilbert space (Sala et al., 2023). This dramatically reduces ergodicity and leads to non-thermal behavior.

D. UV/IR Mixing and Anomalies:

Projecting to the invariant subspace of certain modulated symmetries fails to factorize over sites and produces gravitational anomalies. Holographically, this is resolved by viewing the 1D theory as the boundary of a higher-dimensional topological phase (e.g., generalized toric code), establishing a one-to-one correspondence between bulk electric-magnetic duality and boundary non-invertible duality (Seo et al., 6 Nov 2024).

3. Mathematical Formalism and Classification

Spatially modulated symmetries are not generically onsite and often generate nontrivial algebraic structure when combined with spatial symmetries: $G = G_\text{int} \rtimes G_\text{sp}$ Here, $G_\text{sp}$ acts nontrivially on $G_\text{int}$ via conjugation: $S(g) = S g S^{-1}$, encoding modulation in the symmetry group structure (Bulmash, 8 Aug 2025).

Defect Network Construction (DN):

The defect network recipe for SPT phases protected by modulated symmetries involves assigning local SPT invariants $[\omega]$ on $d$-cells and “anomaly trivialization” cocycles $[\mu]$ on $(d-1)$-cells such that: $S^* \omega/\omega = d\mu$ with invariance constraint $S^*[\omega] = [\omega]$, and lower-dimensional data to cancel residual anomalies.

Cohomological Classification:

Strong SPT invariants must remain fixed under the pullback by the spatial symmetry, restricting the allowed elements of $H^{d+1}(G_\text{int}, U(1))$ (Bulmash, 8 Aug 2025, Han et al., 2023). Weak invariants are decorated on lower-dimensional cells and further modded out by equivalence relations induced by translation or spatial symmetries.

Symmetry Gauging and Non-invertible Duality:

Gauging a finite Abelian modulated symmetry (for prime or composite on-site Hilbert space) involves introducing gauge fields with Gauss law constraints reflecting the modulation. The resulting dual symmetry on gauge fields features a reflected modulation, with an isomorphism implemented by a lattice reflection operator $M$ under appropriate linear algebraic or ring-theoretic conditions (Pace et al., 18 Jun 2024). The duality operator $D$ generically exhibits non-invertible fusion rules: $$D \times D = \mathcal{C} \prod_{q=1}^{n} (1 + U_q + \dots + U_q^{N-1})$$ producing projection operators onto neutral (symmetry-invariant) sectors. The non-invertibility is resolved by embedding the theory on the boundary of an appropriate topological order (Seo et al., 6 Nov 2024).

4. Physical Instantiations: Models and Phenomenology

A. Topological Quantum Chains and SPT Phases

• Dipolar/quadrupolar SPTs: Ground states are constructed via “decorated domain wall” formalism, where domain walls are “colored” by position-dependent charges (linear, quadratic, or exponential spatial dependence) (Han et al., 2023).
• Bundle symmetry: On closed chains, global modulated symmetry generators may not be well-defined; their protection is maintained via bundle structures and transition functions between local patches.
• MPS tensor structure: The presence of modulation alters the projective algebra of edge modes and the virtual symmetry representations in matrix product state descriptions, circumventing cluster model no-go theorems.

B. Fracton and Subsystem Symmetry Phases

• Subsystem and dipole symmetries: Flipping all spins along a row or acting with a spatially modulated operator leads to conservation laws for charge and higher moments, protecting fracton order and constraining quasiparticle motion (Ebisu et al., 25 Sep 2024).
• Non-invertible duality defects: Gauging these symmetries systematically generates fusion rules involving higher-form charges (e.g., lineons in 3D), building a categorical framework for dualities in fractonic and symmetry-enriched topological phases.

C. Quantum Liquids with Modulated Conservation

• One-dimensional rotor and bosonic models with modulated charge conservation display rich phase diagrams: transitions between Mott insulator and quasi-long-range ordered (QLRO) “Bose surface” phases, with c = 2 Luttinger liquid behavior and emergent vortex gases with modulated Coulomb interactions (Sala et al., 2023). For incommensurate modulation, Hilbert space fragmentation impedes ergodicity and drives nontrivial ground-state splitting.

D. Floquet and Space-Time Modulated Symmetries

• In periodically driven (Floquet) systems and electromagnetic media, spatial and temporal modulations intertwine, giving rise to spatio-temporal groups, which are extensions of point and space groups incorporating fractional time translations (Padmanabhan et al., 2017). This impacts selection rules, high-harmonic generation, and the classification of Floquet topological phases.

5. Spatially Modulated Symmetry and Quantum Anomalies

Spatially modulated symmetries modify Lieb-Schultz-Mattis (LSM) anomaly structure in several ways:

• Permuting anyons under translation: In SET orders, lattice translations can permute anyons, leading to foliated field theories and modifications of the continuum SymTFT description (Pace et al., 2 Jul 2025).
• Crystalline Equivalence Principle → Generalization: Classification of SPT phases must enforce invariance of strong data under spatial symmetry pullback; defects in the network may acquire additional 'anomaly' if this is not satisfied (Bulmash, 8 Aug 2025).
• Bulk-boundary correspondence and holography: Non-invertible boundary duality in a 1D modulated clock model reflects the boundary endpoint of a bulk symmetry defect (e.g., in a generalized toric code), where nontrivial modulated symmetry imposes nonlocal constraints leading to UV/IR mixing and emergent gravitational anomalies (Seo et al., 6 Nov 2024).

6. Experimental and Numerical Manifestations

• Emergent density waves and current modulations: Holographic models, such as the Sakai-Sugimoto construction, predict onset of spatially modulated charge/current waves in quark-gluon plasma at sufficiently high baryon density and Chern-Simons coupling (Ooguri et al., 2010, Donos et al., 2011).
• Detection in transport: Modulated nematic order in correlated electron materials yields distinct signatures in transport tensors, offering experimental handles for identifying spatially modulated phases (Kee et al., 2013).
• Domain pattern engineering: In multiferroics, the “flexo-antiferrodistortive” coupling governs domain modulation period and incommensurability, allowing for targeted design of modulated domains and emergent improper ferroelectricity (Eliseev et al., 2013).
• Topological end states: Fermionic chains with modulated interactions display edge-localized fractional charges and nontrivial Berry phase/Chern number structure, signifying topological CDW/Mott phases with fractal quasiparticle spectra (Zuo et al., 2020).

7. Open Directions and Theoretical Challenges

Open questions in the paper of spatially modulated symmetries involve:

• Formal cohomological classification of SPTs and SETs with general spacetime and modulated symmetries, especially for higher dimensions and non-Abelian groups (Pace et al., 2 Jul 2025, Bulmash, 8 Aug 2025).
• Detailed understanding of UV/IR mixing, gravitational and 't Hooft anomalies arising from nonlocal projections associated with modulated symmetry sectors.
• Precise mathematical underpinnings (e.g., via ring theory) for conditions under which modulated and dual symmetries are isomorphic after gauging for composite Hilbert space dimensions (Pace et al., 18 Jun 2024).
• Extending defect network and symmetry-enriched TFT constructions to cases involving time-reversal, reflection, or more general spacetime symmetries intertwined with modulation.
• Engineering synthetic platforms—ultracold atoms, designed metamaterials, superconducting circuits—to realize and diagnose modulated symmetry-protected phases, their defects, and associated anomalous responses.

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Spatially modulated symmetries, by encoding conservation laws at finite momentum, higher moments, or on subdimensional subsystems, fundamentally enrich the classification, dynamics, and anomaly structure of quantum phases. They serve as organizing principles for topological matter, non-invertible dualities, and quantum information, with deep connections to cohomology, field theory, and experimental observables across multiple disciplines.