Multipole Symmetries in Field Theories

• Multipole symmetries are generalized invariance principles that conserve higher-order moments (e.g., dipole, quadrupole) across electromagnetic, gravitational, and quantum systems.
• They emerge from residual gauge freedoms and coordinate transformations, leading to explicit multipole charges and enforcing strict constraints on radiation and dynamics.
• Their algebraic structure underpins novel phenomena in fracton physics and topological phases, guiding the design of quantum materials and symmetry-adapted interactions.

Multipole symmetries are generalized symmetry principles associated with the invariance and conservation of higher-order “multipole moments,” extending beyond mere total charge or mass. In their most developed forms, multipole symmetries appear as residual gauge or coordinate transformations in gauge theories and gravity, as geometric invariants in quantum and classical systems, and as explicit algebraic structures governing conservation laws, constrained dynamics, and emergent phenomena across many branches of theoretical physics. These symmetries enforce nontrivial conservation laws for multipole moments—quantities such as dipole, quadrupole, and higher moments—that impose profound constraints on local and global dynamics, electromagnetic and gravitational radiation, nonergodic quantum phases, and the design of quantum materials.

1. Multipole Symmetries in Gauge Theories

The canonical realization of multipole symmetries in electromagnetism arises from the residual freedom of gauge transformations in Maxwell theory after partial gauge fixing. Explicitly, after imposing Lorenz gauge, residual gauge parameters $\lambda(\mathbf{x})$ must solve the Laplace equation and admit an expansion in spherical harmonics: $$\lambda(\mathbf{x}) = -\sum_{\ell m} \left[ c_{\ell m}^+ r^\ell + c_{\ell m}^- r^{-\ell-1} \right] Y_{\ell m}^*(\theta, \phi)$$ Only the growing modes ($c_{\ell m}^+$) yield nontrivial conserved charges. The associated “multipole charges” are codimension-2 surface integrals,

$$Q_{\ell m} = \oint_{S} d\Sigma_{r} F^{r0} r^\ell Y_{\ell m}^*(\theta, \phi)$$

which, in stationary configurations, become proportional to the classical multipole moments $q_{\ell m}$ of the charge distribution: $Q_{\ell m} = \frac{\ell + 1}{2\ell + 1} q_{\ell m}$ The total multipole charge can be decomposed into a “hard” part from the matter ($q_{\ell m}$) and a “soft” part from the electromagnetic field ($-(\ell/(2\ell+1))q_{\ell m}$), reflecting a precise equipartition between the two. Under time-dependent dynamics (e.g., radiation emission) the difference in hard multipole charges is exactly compensated by the soft charge carried by the outgoing field, leading to exact conservation of the full multipole charge (Seraj, 2016).

In general relativity, analogous concepts hold: “multipole symmetries” are defined as residual diffeomorphisms in canonical harmonic gauge, generating canonical Noether charges whose evaluation yields the full set of mass, current, and momentum multipole moments (Compère et al., 2017, Chakraborty et al., 2021). The associated surface charges coincide with the classical (e.g., Thorne or Geroch-Hansen) multipole moments for compact sources or black holes in both asymptotically flat and de Sitter spacetimes.

2. Multipole Algebra and Conservation Laws

The abstraction of multipole symmetries leads to a structured algebra—the multipole algebra—that generalizes spacetime symmetry. It extends global $U(1)$ (charge) symmetry by including polynomial shift symmetries: $$\delta \phi(x) = \sum_{a, I_a} \lambda_{I_a} P_a^{(I_a)}(x)$$ with $P_a^{(I_a)}(x)$ homogeneous polynomials. The multipole algebra includes generators for translations $T_i$, rotations $R_{ij}$, and multipole shifts $\mathcal{P}_a^{(I_a)}$, closing under commutators such as $[T_i, \mathcal{P}_a^{(j)}]$ and $[R_{ij}, \mathcal{P}_a^{(I_a)}]$ (Gromov, 2018). This algebra ensures the conservation of not just charge,

$Q = \int d^d x\, \rho(x)$

but also all (or a designated set of) higher moments,

$$Q^{(I_a)} = \int d^d x\, \mu_{i_1 \dots i_a}^{(I_a)} x^{i_1}\dots x^{i_a} \rho(x)$$

In systems where the multipole symmetry is maximal (up to degree $n$), every multipole up to $n$ is conserved, directly constraining the allowed dynamics and leading, for example, to fractonic behavior in lattice models.

Gauge field theories in this setting are built by constructing derivative operators $D_\alpha$ annihilating the polynomial shift invariants (i.e., $D_\alpha P = 0$), with the theory’s Lagrangian involving only such invariant derivatives. Gauging only the multipole symmetry (keeping translations/rotations global) leads naturally to symmetric tensor gauge theories ubiquitous in fracton physics and to generalized gauge theories when the symmetry is less maximal (Gromov, 2018). The associated local (Gauss law) constraints enforce conservation of both total and multipole charges.

3. Multipole Symmetries and Constraints on Radiation

The presence of multipole symmetries enforces strict constraints on possible patterns of electromagnetic and gravitational radiation:

• Electromagnetism: For each $(\ell, m)$ multipole, the conservation law

$$\frac{d}{dt} [ Q_{\ell m}^{(h)} + Q_{\ell m}^{(s)} ] = -(\text{flux through the bounding sphere})$$

forces electromagnetic radiation to balance changes in multipole moments of the source with precisely determined radiative multipole “soft” charges. This supplies an infinite hierarchy of constraints governing the radiative emission patterns, independent of explicit dynamical solutions (Seraj, 2016).

• Gravity: In the Noether charge approach, the time variation of the source's multipole moment is equated to the flux of the corresponding conserved Noether (multipole) charge through null infinity, again without reference to the explicit spacetime solution (Compère et al., 2017).

These symmetries underpin “soft memory effects” in which radiation encodes stored information about past source dynamics.

4. Geometric and Quantum Manifestations

Multipole symmetries also admit powerful geometric and group-theoretical realizations:

• Geometric Representation: The Maxwell–Sylvester formalism encodes a rank-$\ell$ multipole as a set of $\ell$ directional vectors on a sphere; physical features, including intrinsic symmetries and angular momentum structures, become manifest as geometric relations among these vectors (Bruno, 2018, Romero et al., 15 Jan 2024). Majorana-star representations similarly relate the geometric distribution of stars to SU(2)-invariant multipole moments of quantum spin states.
• Coordinate-Free Quantum Operators: Multipole operators in quantum systems are most transparently handled using irreducible tensor operator expansions, yielding basis-independent, SU(2)-covariant, and geometrically interpretable objects with direct connection to the system’s symmetry properties.
• Multipole Groups on Lattices: On arbitrary crystal lattices, multipole symmetries are algorithmically classified based on representation theory, lattice translations, and basis site permutations (Bulmash et al., 2023). Emergent subsystem symmetries and vector multipole conservation (without monopole conservation) are possible on non-hypercubic lattices, leading to novel fracton and localization phenomena.

5. Impact on Dynamics, Hydrodynamics, and Symmetry Breaking

Multipole conservation fundamentally alters dynamical relaxation:

• Hydrodynamics: The order of spatial derivatives in hydrodynamic equations increases with the conserved multipole order. For dipole conservation, the continuity equation becomes fourth-order in space,

$$\partial_t \rho(x,t) + B\,\partial_x^4 \rho(x,t) = 0$$

so that relaxation is subdiffusive: the relaxation time diverges as $k^{-4}$ for $k \to 0$. Quadrupole conservation further increases this to sixth order, and so forth (Iaconis et al., 2020).

• Subsystem and Modulated Symmetries: Generalizations to spatially modulated multipole symmetries yield conservation of oscillatory or exponentially localized modes rather than global moments, resulting in novel long-lived edge states or lines/surfaces of conserved momenta in higher dimensions (Sala et al., 2021).
• Symmetry Breaking: The spontaneous breaking of a rank-$a$ multipole symmetry is forbidden in dimensions $d \leq 2(a+1)$ at $T>0$ (generalized Mermin–Wagner theorem), or $d < a+1$ at $T=0$, due to the extreme softness (highly nonlocal) of the corresponding Goldstone modes (Stahl et al., 2021). Disorder further increases the critical dimension for possible order (by generalized Imry–Ma arguments).

6. Applications in Electromagnetic Structures, Condensed Matter, and Topological Phases

Multipole symmetry principles yield predictions and design tools across multiple domains:

• Metasurfaces: Imposing spatial symmetries on the effective material parameters and scattering matrix of a metasurface tightly constrains allowable multipole responses. The emergence of “multipolar extrinsic chirality” enables chiral optical behaviors—even in geometrically achiral structures—when multipolar (e.g., quadrupolar) terms are symmetry allowed (Achouri et al., 2022).
• Multipolar Interactions and Lattice Orderings: In electronic materials, the complete multipole representation organizes charge, spin, toroidal, and “hidden” orders into a symmetry-adapted hierarchy. The presence or absence of specific multipoles, determined by space-group analysis, governs the emergence of unconventional ferroic, nematic, and topologically nontrivial phases (Hayami et al., 14 Mar 2024, Yambe et al., 12 Jun 2025). For example, antisymmetric DM-like multipole interactions, allowed only under specific symmetry conditions, can stabilize exotic multiple-$Q$ states, including triple-$Q$ quadrupolar order in frustrated magnets (Yambe et al., 12 Jun 2025).
• Fracton and Topological Order: Multipole symmetries are the organizing principle of fracton field theories and foliated BF-type topological phases, controlling the ground state degeneracy, mobility restrictions of excitations, and ground state structure dependent on system size and symmetry (Gromov, 2018, Ebisu et al., 2023, Ebisu et al., 19 Jan 2024). In topological insulators and superconductors, multipole indices (constructed from many-body multipole operators and symmetry generators) serve as robust order parameters quantizing bulk-boundary correspondence for edge/corner states (Tada et al., 2023).

7. Group-Theoretical and Algebraic Classification

Systematic classification of multipole-symmetric SPT phases employs group cohomology. In 1D, spatially modulated (multipole) symmetries induce boundary projective representations, whose algebra is classified using H$^2$ and H$^1$ cohomology groups, depending on the order of the multipole symmetry (monopole/dipole/quadrupole, etc.). The projective structure arising from the pull-through of symmetry actions in matrix-product state formalism reflects the underlying hierarchy of multipole symmetry classes and their boundary anomalies (Saito et al., 11 Sep 2025).

The same algebraic structures govern the permissible types of multipole-multipole interactions in Hamiltonians—symmetric (compass-like) and antisymmetric (DM-like)—with the allowed terms dictated by the representations of the site and bond symmetries under the crystallographic point group (Yambe et al., 12 Jun 2025).

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Multipole symmetries, by encoding higher-order conservation laws and transformation properties, serve as a unifying framework for understanding fundamental constraints in gauge theories, gravitational physics, quantum materials, and topological phases. Their implications extend from the most abstract algebraic structures to concrete predictions of dynamic behavior, material response, and topological phenomena. These symmetries enable classification, design, and deep insight into the interplay of locality, symmetry, and dynamics in complex many-body and field-theoretic systems.