Sub-Symmetry-Protected Topology Overview

• Sub-symmetry-protected topology is a framework where symmetries acting on specific subspaces or lower-dimensional subsystems generate robust, localized modes even when global symmetry is broken.
• Model systems in 1D, 2D, and 3D showcase how selective symmetry actions protect edge, corner, and hinge states, using methods like non-Bloch band theory and tensor network diagnostics.
• Classification and experimental investigations of SSPT phases leverage group cohomology, projective representations, and entanglement measures to diagnose topological order under practical perturbations.

Sub-symmetry-protected topology (often abbreviated as SSPT or "sub-SPT") generalizes the notion of symmetry-protected topological (SPT) phases by allowing the relevant symmetry to act only on a proper subspace or a collection of lower-dimensional subsystems within the full Hilbert space. Unlike conventional SPT phases, which require an exact global symmetry connecting the entire system, sub-symmetry or subsystem-protection allows for boundary-localized or subsystem-localized topological invariants that survive even when the full symmetry is broken or partially reduced. These topological features manifest robust boundary or interface modes, quantized response properties, and nontrivial entanglement signatures, which are insensitive to generic local perturbations absent only the specific sub-symmetry protection. Sub-symmetry-protected topology has been theoretically established across dimensions, including 1D insulators and superconductors, 2D cluster and fractal models, and higher-order topological and symmetry-protected scar subspaces, with explicit diagnostic tools and classification schemes grounded in group cohomology, tensor network methods, field theory dualities, and subsystem SymTFT constructions.

1. Foundational Notions: Sub-symmetry, Subspaces, and Subsystem Symmetries

The defining feature of sub-symmetry-protected topology is the replacement of global symmetry with symmetry actions on proper subspaces or strict lower-dimensional subsystems.

• Sub-symmetry and subspaces: Let $P_i$ denote the projector onto a subspace (e.g., a sublattice or a particular sector), and $G$ a symmetry generator. The sub-symmetry action is $G_i = P_i G$, so that $G_i H G_i^{-1} = \pm H P_i$ (Kang et al., 3 Jun 2024, Wang et al., 2022).
• Subsystem symmetries: In higher dimensions, protecting symmetries may act on lines, planes, or fractal manifolds: $G = \bigotimes_\ell G_\ell$ with each $G_\ell$ acting on a $d$-dimensional subsystem of a $D$-dimensional lattice ($d<D$) (Stephen et al., 2019, Devakul et al., 2018, 1803.02369).
• Physical consequences: The protection by sub-symmetry is strictly linked to the subspace or subsystem; boundary or interface-localized states with support entirely inside this subspace cannot be gapped or scattered unless sub-symmetry is broken on that subspace (Wang et al., 2022, Kang et al., 3 Jun 2024, Kang et al., 19 Dec 2025).

These properties contrast with globally protected SPTs, where every boundary state and the entire bulk wavefunction require an unbroken global symmetry for protection.

2. Model Systems and Boundary Phenomena

Sub-symmetry-protected topology has been concretely demonstrated in a variety of models across spatial dimensions and symmetry classes.

1D: SSH and Kitaev Chains

• In the SSH chain, with chiral symmetry $\Sigma_z$, the addition of a sublattice-selective perturbation (acting e.g., only on A-sites) breaks the full symmetry but preserves an "A-sub-symmetry" or "B-sub-symmetry." As a result, only the boundary state localized on the B sublattice remains pinned at zero energy; its polarization remains quantized, while its counterpart on the broken sublattice is shifted away from zero and loses quantization (Wang et al., 2022, Kang et al., 3 Jun 2024, Verma et al., 10 May 2024).
• In the spinful Kitaev (superconducting) chain, a sub-symmetry-protecting perturbation preserves Majorana zero modes on only one edge via a subspace-projected particle-hole symmetry; the topological invariant (Z or Z$_2$ winding) persists only in the protected subspace (Kang et al., 3 Jun 2024).
• Non-Bloch band theory and the use of the generalized Brillouin zone (GBZ) provide a quantitative framework to compute the correct bulk–boundary correspondence and winding numbers in the presence of sub-symmetry-protection, where the usual Bloch invariants break down (Verma et al., 10 May 2024).

2D: Cluster Models, HOTIs, and Fractal SSPTs

• In the 2D cluster model (rotated square lattice), rigid line symmetries (acting on rows or columns) define the subsystem protection. Edge and corner states transform projectively under those symmetries and display robustness exclusively when the subsystem symmetries are unbroken (Stephen et al., 2019, Devakul et al., 2018, 1803.02369).
• Higher-order topological insulators such as the Benalcazar–Bernevig–Hughes model with onsite potential acting only on one sublattice realize sub-symmetry protection at the corner or hinge: only those modes with support fully on the invariant sublattice remain topological; their state-resolved quadrupole moment remains quantized (Kang et al., 19 Dec 2025).
• Sub-symmetry protection extends to fractal symmetries, requiring specific configurations of local clock operators and exhibiting a rich structure of local projective representations at edges and corners. Nontrivial SSPTs are only possible with sufficiently many independent fractal symmetries (Devakul, 2018).

3D: Planar-Subsystem SSPTs and Enrichment

• In 3D, SSPT phases protected by planar symmetries (e.g., cluster state models on bcc/fcc lattices) exhibit robust boundary, hinge, or loop-localized modes transforming projectively under their respective subsymmetries. These models realize nontrivial corrections to the entanglement area law reflecting their subsystem topology (Stephen et al., 2019, Stephen et al., 2020).
• Gauging of subsystem symmetries yields fracton topological orders, and boundary anomalies (mixed between global and subsystem symmetries) indicate the presence of SSPT order (Stephen et al., 2020).

Weak and Non-Invertible Sub-symmetry-Protected Phases

• Recent lattice constructions exhibit subsystem/weak non-invertible SPT order, where boundary or interface (edge, corner, or hinge) modes are protected by non-invertible subsystem actions, sometimes in conjunction with lattice translation. These yield interface degeneracies and LSM-type anomalies not realizable with ordinary global symmetries (Furukawa, 16 May 2025).

3. Classification and Invariants

Systematic classification of sub-symmetry-protected topology relies on group cohomology, projective representation theory, tensor network analysis, and holographic field-theory dualities.

• Group cohomology: For 2D strong SSPTs with linear subsystem symmetries (e.g., $\mathbb{Z}_2$ per row/column), equivalence classes are labeled by

$$\mathcal{C}[G_s] = H^2[G_s^2, U(1)] / \big(H^2[G_s, U(1)]\big)^3,$$

where $G_s$ is the onsite symmetry group (Devakul et al., 2018, Jia et al., 28 May 2025).

• Projective representation at corners/edges: Nontrivial SSPTs are characterized by projective boundary representations of the subsystem symmetry group. Bulk invariants can be directly computed as commutators of truncated symmetry operators with corner operators (Devakul et al., 2018, Devakul, 2018).
• Entanglement invariants: SSPT order manifests as a constant, universal “symmetry-protected entanglement entropy” (SPEE) correction $\gamma_{\text{SSPT}}$ to the leading area law for the entanglement entropy:

$$S(A) = \alpha |\partial A| - \gamma_{\text{SSPT}} + \dots$$

(Stephen et al., 2019, Stephen et al., 2020). This correction is invariant under arbitrary symmetric local unitaries and differentiates strong subsystem phases from weak ones, which become trivial under linearly-symmetric circuits.

• Field-theory duality and SymTFT: SSPT classifications can be reformulated using subsystem symmetry topological field theory—e.g., 2-foliated BF theory in 3+1D—with different boundary conditions corresponding to trivial, SSB, or nontrivial subsystem SPT phases, directly reproducing the cohomological classification (Jia et al., 28 May 2025).
• Higher-order and scar subspace SPTs: For higher-order and excited-state protected topology, cohomological indicators and long-range string order, projective action of time-reversal/inversion, and nontrivial boundary-twist phases serve as robust invariants (Kang et al., 19 Dec 2025, Matsui et al., 12 Dec 2025).

4. Diagnostics and Experimental Probes

Several distinct signatures and diagnostic procedures confirm the presence of sub-symmetry-protected topology:

• Boundary state robustness: Edge/corner/hinge modes survive under generic perturbations that preserve only the relevant sub-symmetry, but are destroyed by breaking that sub-symmetry—even when all bulk invariants (e.g., quantized Zak or quadrupole phase) are lost (Wang et al., 2022, Kang et al., 19 Dec 2025).
• Entanglement features: A nontrivial constant correction to the entanglement entropy area law—extractable numerically with tensor network approaches—diagnoses SSPT order (Stephen et al., 2019).
• Non-Bloch invariants: In 1D, non-Bloch winding numbers (defined using the generalized Brillouin zone) correctly predict which edge states persist in the presence of sub-symmetry, when ordinary Bloch invariants fail (Verma et al., 10 May 2024).
• State-local polarization, quadrupole moments, and spin texture: Quantized state-resolved local invariants (polarization for 1D, quadrupole for 2D, spin winding for sub-chiral symmetry bands) diagnose the sub-symmetry-protected states (Kang et al., 3 Jun 2024, Kang et al., 19 Dec 2025, Mo et al., 2023).
• Projective representations and string order: Persistent projective string-order parameters (e.g., $O_\text{string} = -4/9$ in the AKLT scar subspace) and robust, symmetry-protected boundary-twist phases provide stringent, experimentally accessible signatures (Matsui et al., 12 Dec 2025).
• Photonic, cold atom, and superconducting platforms: Sub-symmetry-protected topology has been realized and probed in reconfigurable photonic lattices, through site-selective waveguide writing and controlled perturbations, as well as proposed for implementation in semiconductor-superconductor nanowires with designed pinning of Majorana boundary modes (Wang et al., 2022, Kang et al., 3 Jun 2024).

5. Connections to Topological Order, Fractons, and Anomalies

Subsystem and sub-symmetry protection interpolates between conventional SPTs and topological order, with rich links to fracton physics and anomalies:

• Duality to topological order: Strong SSPT phases in 2D map exactly, via duality, to toric code models enriched with nontrivial constraints, with membrane order parameters and boundary zero modes directly mapping to anyons and ground-state degeneracy (Feng, 2023).
• Gauging subsystem symmetries: Promoting rigid subsystem symmetries to local gauge constraints yields fracton orders (e.g., X-cube models), with the nature of flux excitations and ground-state degeneracy reflecting the underlying SSPT structure (1803.02369, Stephen et al., 2020).
• Boundary anomalies: Mixed anomalies at boundaries (e.g., between global and subsystem symmetries) enforce gapless or spontaneously broken boundary phases, realized concretely in the decorated 3D toric code/cluster state models and field-theory constructions (Stephen et al., 2020, Jia et al., 28 May 2025).
• Weak and non-invertible SPTs: Weak SPTs with subsystem or non-invertible protection yield interface anomalies—such as Lieb-Schultz-Mattis–type degeneracies tied to symmetry and translation—which are absent in any model with only local onsite symmetries (Furukawa, 16 May 2025).

6. Higher-Order, Momentum-Selective, and Scar Subspace Generalizations

Recent developments extend the scope of sub-symmetry-protected topology to previously unexplored domains:

• Higher-order systems: Sub-symmetry protection in higher-order topological insulators and semimetals ensures only certain corner or hinge states retain quantized spectral and local invariants when appropriate sub-sublattice or subspace symmetries survive. State-resolved quadrupole moments provide a local diagnostic for which modes remain protected (Kang et al., 19 Dec 2025).
• Sub-chiral and momentum-selective symmetry: Sub-chiral symmetry—momentum-dependent generalization of chiral symmetry—protects boundary flat bands exhibiting k-dependent spin textures and quantized boundary Berry phases, even when all global chiral symmetry is destroyed (Mo et al., 2023).
• Scar subspaces: Dynamically isolated subspaces (e.g., towers of quantum many-body scar states in the AKLT model) can carry and robustly inherit the projective SPT structure, showing sectorwise robust topological signatures (long-range string order, symmetry-twist responses) even in the absence of global symmetry in individual states (Matsui et al., 12 Dec 2025).

7. Outlook: Unified Structure and Future Directions

Sub-symmetry-protected topology unifies diverse topological phenomena under the organizing principle that protection does not require a symmetry to act globally but only on particular subspaces/subsystems. This principle yields robust, quantized, and physically accessible signatures—entanglement, state-local polarization/quadrupole, projective structure at boundaries, and degeneracies at higher-codimension interfaces—that persist in a broad class of models and physical realizations, including those with non-invertible or fractal symmetries. The duality to topological order and connections to field-theory/holographic constructions through SymTFT further solidify the topological and physical foundations of this domain.

Open challenges remain in classifying and experimentally diagnosing sub-symmetry-protected phases for arbitrary subsystem geometries, non-invertible categorical symmetries, higher dimensions, and dynamical or non-thermalized subspaces. Ongoing development of tensor network, field theory, and boundary-anomaly methodologies—alongside experimental realizations in photonics, ultracold atoms, and quantum circuits—continues to drive the field forward (Stephen et al., 2019, Wang et al., 2022, Kang et al., 19 Dec 2025, Jia et al., 28 May 2025, Matsui et al., 12 Dec 2025, Furukawa, 16 May 2025).