Modulated Symmetries in Lattice Systems

• Modulated symmetries are internal symmetries with spatially-varying actions that interact with translation invariance to produce unique anomaly structures and phase behaviors.
• They are realized in lattice systems through site-dependent generators, enabling dipolar, quadrupolar, and exponential operations with distinct boundary and defect effects.
• Their study informs crystalline SPT phases and defect network frameworks, offering actionable insights into symmetry protection and anomalous transport mechanisms.

Modulated symmetries are internal symmetries whose action is spatially non-uniform and, equivalently in much of the recent literature, internal symmetries that are not invariant under spacetime symmetry actions. In lattice systems their generators are site-dependent, translation acts on them nontrivially, and the full symmetry is naturally expressed as a semidirect product such as $G_{\mathrm{int}} \rtimes G_{\mathrm{sp}}$ or $G \rtimes_\varphi \mathbb{Z}_L$. Dipole, multipole, exponential, harmonic, and subsystem symmetries are the standard examples. Current work places them at the intersection of SPT classification, Lieb-Schultz-Mattis anomalies, gauging, symmetry topological field theory, non-invertible operators, open-system topology, and anomalous transport or relaxation (Pace et al., 2 Jul 2025, Yao, 4 Oct 2025, Pace et al., 2024).

1. Algebraic definition and translational covariance

In one-dimensional lattice formulations, modulated symmetry generators are written as

$U_q = \prod_j (X_j)^{f_j^{(q)}},$

where $X_j$ are local symmetry operators and $f_j^{(q)}$ are site-dependent exponents. Translation does not generally commute with these generators, but instead acts by shifting the modulation profile,

$T^m U_q T^{-m} = \prod_j (X_j)^{f_{j-m}^{(q)}}.$

This is encoded by a homomorphism $\varphi : \mathbb{Z}_L \to \mathrm{Aut}(G)$, so that the physical symmetry takes the semidirect-product form $G \rtimes_\varphi \mathbb{Z}_L$. Equivalent formulations use $G_{\mathrm{int}} \rtimes G_{\mathrm{sp}}$, emphasizing that the modulation is an action of spatial symmetry on the internal symmetry data rather than an independent internal structure (Pace et al., 2 Jul 2025, Pace et al., 2024).

A more general translationally covariant continuum criterion is given by the notion of a translationally covariant modulated symmetry. For a charge vector $\vec Q$, translation compatibility is characterized by

$G \rtimes_\varphi \mathbb{Z}_L$0

with $G \rtimes_\varphi \mathbb{Z}_L$1 real commuting matrices. The general solution has the form

$G \rtimes_\varphi \mathbb{Z}_L$2

In one dimension, real Jordan normal form reduces the allowed Abelian modulations to three basic types: multipole or polynomial components, exponential components, and harmonic or oscillatory components. This classification makes precise which spatially varying charges can coexist with translation-invariant Hamiltonians (Chen et al., 6 Jun 2026).

This algebraic viewpoint also clarifies a recurrent theme in the literature: modulated symmetries become physically nontrivial only when spatial symmetry is retained. If translation or other crystalline symmetry is explicitly broken, the modulation can often be removed by blocking or redefinition, whereas with spatial symmetry enforced it produces distinct symmetry actions, distinct anomaly structures, and distinct phase classifications (Bulmash, 8 Aug 2025, Ning et al., 19 Mar 2026).

2. Canonical lattice realizations

The most widely used realizations occur in one-dimensional spin chains. Conventional uniform symmetry is represented by

$G \rtimes_\varphi \mathbb{Z}_L$3

Its dipolar deformation is

$G \rtimes_\varphi \mathbb{Z}_L$4

with translation algebra

$G \rtimes_\varphi \mathbb{Z}_L$5

Higher modulations include quadrupolar generators

$G \rtimes_\varphi \mathbb{Z}_L$6

and exponential generators

$G \rtimes_\varphi \mathbb{Z}_L$7

Exactly solvable Hamiltonians are obtained by generalizing decorated domain walls to modulated defects; for example, a dipolar stabilizer is

$G \rtimes_\varphi \mathbb{Z}_L$8

These models already display several characteristic features of the subject: nontrivial translation algebra, edge projective structure, and sensitivity to boundary conditions (Han et al., 2023).

A more general framework treats a conserved quantity as

$G \rtimes_\varphi \mathbb{Z}_L$9

with arbitrary spatial profile $U_q = \prod_j (X_j)^{f_j^{(q)}},$0. In one-dimensional stochastic cellular automata and related models, the coefficients satisfy linear recursions such as

$U_q = \prod_j (X_j)^{f_j^{(q)}},$1

whose characteristic roots determine whether the symmetry is periodic or quasi-periodic, exponentially localized, or polynomial. In higher dimensions, the conserved modes are described by momentum-space constraints $U_q = \prod_j (X_j)^{f_j^{(q)}},$2, producing lines, loops, or surfaces of conserved momenta rather than isolated $U_q = \prod_j (X_j)^{f_j^{(q)}},$3 modes (Sala et al., 2021).

Bosonic and rotor realizations provide a complementary continuous-symmetry setting. A representative Hamiltonian is

$U_q = \prod_j (X_j)^{f_j^{(q)}},$4

with site-dependent $U_q = \prod_j (X_j)^{f_j^{(q)}},$5 transformations constrained by

$U_q = \prod_j (X_j)^{f_j^{(q)}},$6

When the solutions are $U_q = \prod_j (X_j)^{f_j^{(q)}},$7 and $U_q = \prod_j (X_j)^{f_j^{(q)}},$8, the conserved quantities are finite-momentum Fourier charges,

$U_q = \prod_j (X_j)^{f_j^{(q)}},$9

These models conserve finite-wavevector density components rather than total particle number (Sala et al., 2023).

A distinct two-dimensional realization occurs in $X_j$0 topological ordered phases with commuting projectors of odd support $X_j$1 along the horizontal direction. The modulated symmetry data $X_j$2 satisfy

$X_j$3

The symmetry group is $X_j$4, the anyons can move only in rigid steps of size $X_j$5, and the ground-state degeneracy depends on lattice size as

$X_j$6

This lattice-size dependence is identified as a manifestation of ultraviolet/infrared mixing (Yoshitome et al., 12 Jun 2025).

3. Modulated SPT phases and classification schemes

One line of development generalizes decorated domain wall constructions to modulated symmetry defects. In one-dimensional chains this yields modulated SPT phases protected by dipolar, quadrupolar, and exponential generators. A salient feature is that modulated symmetries are generically only present for open chains and are broken upon imposing periodic boundary conditions. Nevertheless, SPT order can persist with periodic boundary conditions through what is termed “bundle symmetry”: symmetry operators are well defined on local patches and patch together through transition functions, analogously to a nontrivial fiber bundle (Han et al., 2023).

The defect-network approach extends the crystalline equivalence principle to modulated symmetry. In this framework, modulated symmetries can be treated identically to unmodulated symmetries in the absence of spatial symmetries, but with spatial symmetry present some defect networks that are non-anomalous for unmodulated symmetries become anomalous. For strong SPT data, anomaly-freeness requires

$X_j$7

for all spatial symmetry elements $X_j$8. In $X_j$9D, weak SPT data for translation is classified by the coinvariants

$f_j^{(q)}$0

This gives a real-space formulation of how modulation restricts admissible defect decorations and admissible crystalline SPT data (Bulmash, 8 Aug 2025).

Matrix-product-state methods refine this picture. For modulated symmetry, the standard symmetry push-through condition is replaced by

$f_j^{(q)}$1

with site-dependent virtual operators, and translation covariance imposes

$f_j^{(q)}$2

Projective virtual symmetries satisfy [ v_j(g)v_j(h)=\omega_j(g,h)\,v_j(gh