Non-Invertible SPT Phases

• Non-invertible SPT phases are quantum states characterized by fusion-category symmetries, exhibiting short-range entanglement and robust edge or interface modes.
• They are realized in lattice and spin models using tensor networks, duality transformations, and fiber functor classifications to yield distinct phase structures.
• Their protected boundary phenomena, including anomalous degeneracies and corner modes, offer new experimental and theoretical avenues for quantum device development.

A non-invertible symmetry protected topological (SPT) phase is a quantum phase of matter, protected by a non-invertible symmetry, that exhibits robust edge or interface phenomena and short-range entanglement. Unlike conventional (invertible or group-based) SPTs, which are classified by group cohomology and possess stacking group structure, non-invertible SPTs are protected by categorical symmetries—namely, fusion categories—lacking group inverses and subgroup structures. Such phases are realized in lattice and spin models, tensor-network approaches, and field theoretic frameworks, and are associated with new algebraic structures, protected degeneracies, exotic boundary behavior, distinct topological orders, and robust dynamical signatures.

1. Algebraic Structure and Symmetry Framework

Non-invertible SPT phases are characterized by symmetries implemented not by groups but by fusion categories or related higher categorical objects (often denoted 𝒞). In this framework, “symmetry operators” correspond to topological defect lines or non-invertible MPOs (matrix product operators) rather than global on-site group actions. The absence of invertibility is encoded in fusion rules such as

$D^2 = 1 + \eta^e + \eta^o + \eta^e\eta^o$

as for the $\mathrm{Rep}(D_8)$ symmetry in the cluster model, where $D$ is a duality operator and $\eta^{e/o}$ are $\mathbb{Z}_2$ generators (Seifnashri et al., 1 Apr 2024). Unlike groups, fusion categories may have simple objects (symmetry lines) with quantum dimensions greater than unity and non-invertible fusion coefficients $N_{ab}^c$.

A central mathematical structure is the fiber functor $f : \mathcal{C} \rightarrow \mathrm{Hilb}$, which specifies how the categorical symmetry is realized locally (on-site) on the physical Hilbert space. For lattice models to truly realize an “anomaly-free” fusion category symmetry—and hence admit a trivial SPT phase—such a fiber functor is necessary (Meng et al., 29 Dec 2024). The classification of non-invertible SPTs is closely tied to the possible inequivalent fiber functors of a given fusion category.

2. Lattice Realizations and Model Construction

Non-invertible SPT phases have been realized in various spin systems and cluster-type models. In $(1+1)$d, the canonical example is the cluster model with $\mathbb{Z}_2 \times \mathbb{Z}_2$ on-site symmetry and an emergent non-invertible symmetry described by the fusion category $\mathrm{Rep}(D_8)$ (Seifnashri et al., 1 Apr 2024). The Hamiltonian is

$H_{\text{cluster}} = -\sum_j Z_{j-1} X_j Z_{j+1}$

with $X_j$, $Z_j$ Pauli operators, and non-invertible operator $D$ acting via:

$$D X_j = (Z_{j-1} Z_{j+1}) D, \qquad D^2 = 1 + \eta^e + \eta^o + \eta^e\eta^o.$$

Additional exactly solvable Hamiltonians corresponding to inequivalent $\mathrm{Rep}(D_8)$-fiber functors have been constructed, yielding three distinct non-invertible SPT phases classified by categorical data (Meng et al., 29 Dec 2024).

General constructions utilize twisted gauging and duality transformations (e.g., the Kennedy–Tasaki (KT) transformation), mapping between SPT phases and symmetry-broken phases. For instance, starting from an anomalous abelian symmetry in a $(2+1)$d bulk SET phase, gauging a subgroup and applying partial dualities can yield $(1+1)$d boundaries with non-invertible $\mathrm{Rep}(G)$ symmetry (Li et al., 24 May 2024). Such constructions have been systematically extended to $\mathrm{Rep}(Q_8)$ and related categories.

Subsystem and higher-order SPTs protected by non-invertible symmetries are constructed on generalized lattice geometries (e.g., bipartite cubical graphs $\mathbb{C}(d,0,q)$) with non-invertible symmetry operators acting on high-codimension regions, leading to protected interface or corner modes (Furukawa, 16 May 2025, Mana et al., 23 May 2025).

3. Edge Modes, Interfaces, and Bulk-Boundary Correspondence

A signature of non-invertible SPT phases is the existence of symmetry-protected boundary or interface modes. While ordinary SPTs exhibit projective representations of the symmetry group at the edge, non-invertible SPTs in the $\mathbb{Z}_2\times\mathbb{Z}_2$ cluster chain, for example, exhibit edge modes whose algebra closes only up to the fusion rules of the categorical symmetry:

$$\eta^{o}_{\text{local}} D_{\text{local}} = -D_{\text{local}} \eta^{o}_{\text{local}}.$$

This structure persists even under local, symmetry-preserving perturbations.

Interface modes between distinct non-invertible SPTs manifest as protected degeneracies, which can be detected via the representation theory of the “interface algebra” obtained from the MPOs acting on the matrix product state (MPS) boundary (Inamura et al., 28 Aug 2024). When joining two different SPT phases, the interface operator algebra always lacks one-dimensional representations, enforcing anomalous, degenerate edge states.

In higher-order topological phases, these edge phenomena generalize to corner or hinge modes protected solely by non-invertible symmetry. For instance, robust Majorana-like corner modes arise at the intersection of domain walls separating distinct non-invertible cluster-type SPTs in $(2+1)$d and $(3+1)$d (Mana et al., 23 May 2025).

4. Classification Schemes and Topological Invariants

The classification of non-invertible SPT phases requires new invariants beyond group cohomology. In the categorical language, phases correspond to Q-systems (special Frobenius algebras) in the charge category associated to the fusion category symmetry, subject to the “matrix algebra” criterion upon forgetting the symmetry (Meng et al., 29 Dec 2024). Each SPT phase is uniquely labeled by the fiber functor up to monoidal equivalence.

Hierarchical Hasse diagrams organize both gapped and gapless SPT/SSB phases as condensable algebras in the Drinfeld center $Z(\mathrm{Rep}(D_8))$ or similar centers (Bhardwaj et al., 1 Mar 2024). Gapped SPTs correspond to Lagrangian algebras; gapless analogues correspond to non-maximal condensations. The IR content is fully characterized by pivotal tensor functors from the symmetry category to module (or multi-module) categories:

$\varphi : \mathrm{Rep}(D_8) \to \mathcal{C}$

where $\mathcal{C}$ is, for example, $\text{Vec}$ or a matrix-category for gSSB phases.

For parameterized families of SPTs, non-abelian Thouless pumps are classified by non-abelian invariants in the automorphism group of the fiber functor, generalizing conventional cohomological invariants (Inamura et al., 28 Aug 2024).

In certain settings, distinct non-invertible SPT phases cannot even be connected by a symmetric entangler (finite-depth, globally symmetric circuit), reflecting the intrinsic absence of the stacking operation that typifies invertible SPTs (Seifnashri et al., 1 Apr 2024). However, for some categories, such as $\mathrm{Rep}(A_4)$, it is shown that a symmetric entangler exists precisely when the phases are related by a “fixed-charge duality”, and it can be constructed as a matrix product unitary (You, 4 Sep 2025).

5. Physical Properties and Dynamical Signatures

Non-invertible SPTs display robust physical phenomena tied to the categorical symmetry. The degeneracies associated with the non-invertible fusion algebra remain protected against local perturbations that respect the non-invertible symmetry, even in the presence of disorder or out-of-equilibrium driving (Li et al., 19 Aug 2025). For example, in disordered Hamiltonians with Rep$(D_8)$ symmetry, edge modes persist as exact zero modes at zero temperature and exhibit slow oscillations or period-doubled Floquet dynamics when periodic driving is used.

Experimentally, interface and edge phenomena of non-invertible SPTs have been demonstrated using photonic waveguide lattices, where “sub-symmetry” protection of edge or corner states remains even after the bulk SPT invariant is destroyed (Wang et al., 2022).

In extended models, such as modulated SPT (MSPT) phases protected by dipolar or exponentially modulated charges, non-invertible Kramers-Wannier and Kennedy-Tasaki dualities provide bulk diagnostics and duality-connected SSB phases, testable via edge projective analyses and “topological-holographic” correspondences to bulk topological orders (Kim et al., 3 Jul 2025).

6. Interfaces, Anomalies, and Weak/Subsystem SPT Phenomena

Non-invertible SPT phases can also be constructed with subsystem or weak symmetry, leading to exotic interface behavior that extends the notion of boundary protection to higher codimension (e.g., corners, hinges). In such models, the symmetry acts non-invertibly on codimension-$>1$ submanifolds, so interface modes appear at corners or hinges and are protected by a projective algebra (Furukawa, 16 May 2025, Mana et al., 23 May 2025).

When translation symmetry is combined with non-invertible symmetry in weak SPT settings, the resulting interface models may exhibit “exotic Lieb-Schultz-Mattis anomalies”: the effective boundary algebra prevents the existence of a trivial gapped ground state, forcing protected degeneracies or gapless behavior at edges or hinges.

7. Methodological and Theoretical Implications

The paper and classification of non-invertible SPTs unify techniques from tensor network theory (injective MPS/MPO with categorical symmetries), lattice model engineering (with duality mappings and subsystem symmetries), topological field theory (SymTFT, Drinfeld centers, Q-systems), and bulk-boundary holography (topological-holographic correspondence, anomaly inflow). Duality frameworks map the classification problem for topological phases with non-invertible symmetry to those with conventional group symmetry, but reveal fundamentally new phase structures and protected phenomena not accessible by group cohomology (Cao et al., 27 Feb 2025).

Non-invertible symmetries thus broaden the landscape of SPT phases, revealing robust, new types of topological protection, anomalous interface physics, and unique categorical structures, and extending the classification and realization of SPT orders beyond existing paradigms.