⚛️ high-energy theory

Generalized cluster states in 2+1d: non-invertible symmetries, interfaces, and parameterized families

This paper constructs and analyzes 2+1-dimensional lattice models of symmetry-protected topological phases with non-invertible symmetries, known as generalized cluster states, by gauging subgroup symmetries and utilizing tensor networks to demonstrate that their interfaces are described by strip 2-algebras leading to degenerate modes and to establish a framework for generalized Thouless pumps.

Original authors: Kansei Inamura, Shuhei Ohyama

Published 2026-08-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kansei Inamura, Shuhei Ohyama

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, matter organizes itself into distinct phases, much like water freezing into ice or boiling into steam. However, some of these phases are far stranger than the everyday states we know. They are called symmetry-protected topological phases. In these states, the material looks ordinary on the inside, but its edges or boundaries hold a secret, protected by the underlying symmetries of the system. If you try to smooth out the edge or remove the protection, the material resists, often forcing the edge to become disordered or "degenerate," meaning it cannot settle into a single, calm state. This connection between the hidden interior and the active edge is a fundamental rule of modern physics. Recently, scientists have begun to explore what happens when the rules of symmetry themselves change. Instead of the usual, reversible symmetries found in nature, they are investigating "non-invertible" symmetries. These are more complex rules where actions cannot simply be undone, creating a richer, more intricate landscape of possibilities for how matter can behave.

A team of researchers has now built a detailed map of this new landscape, constructing specific models of these exotic quantum states in three dimensions—two spatial dimensions plus time. They focused on a class of materials they call generalized cluster states. To create these, they started with a standard quantum system and performed a mathematical operation known as "gauging," which essentially turns a global symmetry into a local one, weaving the symmetry into the fabric of the material itself. This process generates a new type of order protected by non-invertible symmetries. The researchers did not just write down equations; they constructed the actual quantum states using a powerful tool called tensor networks. You can think of a tensor network as a way of describing a complex quantum state by breaking it down into a grid of interconnected pieces, similar to how a high-resolution image is made of pixels, but here the connections carry the quantum information. This method allowed them to see the microscopic structure of these states clearly, revealing how the non-invertible symmetries act on the material.

One of the most significant discoveries concerns what happens when two of these different quantum phases meet. In ordinary physics, if you bring two different materials together, the boundary between them might be smooth or it might have a specific pattern. However, the researchers found that when two generalized cluster states belonging to different phases meet, the boundary is forced to be degenerate. This means the interface cannot settle into a single, unique ground state. It is as if the boundary is stuck in a state of perpetual uncertainty, unable to choose one configuration over another. This happens regardless of whether the boundary is a sharp, clean line or a messy, gapless region. The symmetry itself enforces this disorder. This finding generalizes a well-known principle in physics called the bulk-boundary correspondence, extending it to these new, non-invertible symmetries. The researchers showed that this degeneracy is a direct consequence of the mathematical structure of the symmetry at the interface, which they described using a new algebraic framework they call a "strip 2-algebra."

The team also explored how these states behave when they are slowly changed over time. They created families of these quantum states that could be smoothly tuned, like turning a dial, while keeping the symmetry intact. By moving this dial in a circle, they demonstrated a phenomenon known as a generalized Thouless pump. As the system cycles through the parameter space, it pumps a specific type of excitation from one side to the other. This is not just a theoretical curiosity; it confirms that these new phases of matter have a non-trivial topological structure that can transport information or charge in a quantized way. The researchers tested their ideas on concrete examples, including cases involving simple groups like the integers modulo two, and found that their general predictions held true. They compared their results with known phases and showed that their construction captures the essential features of these complex systems.

Ultimately, this work provides a systematic way to understand and construct these exotic quantum phases. By using tensor networks, the researchers were able to visualize the symmetry operators and the interface modes, turning abstract mathematical concepts into concrete, calculable objects. They proved that the interface between different phases must be degenerate, a result that holds even when the interface is gapless, which is a stronger constraint than what is seen in systems with ordinary symmetries. This suggests that non-invertible symmetries impose a stricter order on the quantum world, forbidding certain smooth transitions that would otherwise be possible. The study also clarified the classification of these phases, showing how they are labeled by specific mathematical choices and how they relate to one another. By building these models and studying their properties, the researchers have opened a window into a new realm of quantum matter, where the rules of symmetry are more complex, and the behavior of the material at its edges is fundamentally constrained by the hidden algebra of the universe.

Technical Summary: Generalized Cluster States in 2+1d

Problem Statement

Symmetry-protected topological (SPT) phases are non-trivial phases of quantum matter distinguished by their symmetry constraints and protected edge modes. While the bulk-boundary correspondence and topological pumping (generalized Thouless pumps) are well-established for systems with invertible symmetries (groups), recent developments have extended the concept of symmetry to include non-invertible symmetries, described algebraically by fusion categories and higher categories.

A significant gap exists in the systematic understanding of 2+1-dimensional SPT phases protected by non-invertible symmetries, specifically those described by fusion 2-categories. While lattice models for such phases have been constructed (e.g., via gauging finite group symmetries), there is no unified framework describing:

The explicit tensor network representations of their ground states and symmetry operators.
The nature of the symmetry acting on interfaces between different SPT phases (specifically, whether the interface is degenerate).
The behavior of these phases under parameterized families (topological pumping) in the presence of non-invertible symmetries.

Methodology

The authors employ a combination of lattice gauge theory, tensor network states (specifically Projected Entangled Pair States - PEPS), and higher category theory to construct and analyze 2+1d SPT phases.

Construction via Gauging: The authors construct lattice models by starting with a system possessing a finite group $G$ symmetry that spontaneously breaks to a subgroup $K$ . They then gauge a subgroup $H \subset G$ . The resulting models realize SPT phases protected by a group-theoretical fusion 2-category symmetry, denoted $\mathcal{C}(G; H)$ .
Generalized Cluster States: The ground states of these models are identified as "generalized cluster states." The authors provide explicit PEPS representations for these states, where the local tensors encode the group structure and the gauging procedure.
Tensor Network Symmetry Operators: Symmetry operators corresponding to objects in the fusion 2-category are constructed using Projected Entangled Pair Operators (PEPOs). These are built from gauging ( $D_H$ ) and ungauging ( $\bar{D}_H$ ) operators acting on the underlying $G$ -symmetric model. The authors demonstrate that these operators act "fractionally" on the virtual bonds of the PEPS.
Interface Analysis:
- Category-Theoretic: Interfaces between two SPT phases (labeled by complements $K_1$ and $K_2$ ) are described by the strip 2-algebra (a multifusion category) $\mathcal{C}_{K_1, K_2}$ . The authors analyze the module categories over this algebra to classify gapped interface modes.
- Tensor Network: The symmetry action on the interface is explicitly computed using the fractionalized action tensors derived from the bulk PEPS.
Parameterized Families: Using the symmetry interpolation method, the authors construct $S^1$ -parameterized families of these states. They analyze the adiabatic evolution to demonstrate the pumping of non-trivial interface modes (generalized Thouless pumps).

Key Contributions and Results

1. Construction of Generalized Cluster Models

The paper defines a class of exactly solvable lattice models in 2+1d. By gauging a subgroup $H$ in a $G$ -symmetric model broken to $K$ (where $HK=G$ and $H \cap K = \{e\}$ ), the authors realize SPT phases with symmetry $\mathcal{C}(G; H)$ .

Result: The ground states are unique and can be written as equal-weight superpositions of configurations, generalizing the 1+1d cluster state.
Representation: Explicit PEPS representations are provided, with virtual bonds labeled by elements of $H$ .

2. Non-Invertible Symmetry Operators

The authors explicitly construct the symmetry operators for the fusion 2-category $\mathcal{C}(G; H)$ .

Result: The 0-form symmetry operators are given by $D[V^g_H] = D_H U_g \bar{D}_H$ , where $U_g$ is the on-site symmetry of the ungauged model.
Fractionalization: It is shown that these operators act on the virtual legs of the PEPS via "action tensors" $\phi_K[V^g_H]$ . This fractionalization is a key feature allowing the study of boundary and interface physics.
Bond Dimension Reduction: For non-anomalous symmetries, the bond dimension of the PEPO representing the symmetry can be reduced from $|H|^2$ to $|H|$ , analogous to "onsiteability" in 1+1d.

3. Interface Symmetries and the Strip 2-Algebra

A central result concerns the symmetry structure at the interface between two generalized cluster states (potentially in different SPT phases).

Result: The symmetry at the interface is described by a strip 2-algebra (a multifusion category), denoted $\mathcal{C}_{K_1, K_2}$ .
Degeneracy Theorem: The authors prove that if the two SPT phases are distinct (i.e., their defining fiber 2-functors are not isomorphic), the interface symmetry is anomalous. Consequently, any state at the interface must be degenerate (symmetry-enforced degeneracy), regardless of whether the interface is gapped or gapless.
Generalization: This generalizes the bulk-boundary correspondence. Unlike invertible symmetries where gapless edges can be non-degenerate, non-invertible symmetries enforce degeneracy on interfaces between distinct phases.

4. Generalized Thouless Pumps

The authors construct $S^1$ -parameterized families of these generalized cluster states.

Method: They interpolate between the identity and a domain-wall creation operator (which permutes symmetry-breaking ground states) while preserving the non-invertible symmetry.
Result: Adiabatic evolution along the parameter space $S^1$ pumps a non-trivial self-interface mode. This mode corresponds to a non-trivial representation of the strip 2-algebra, establishing the existence of generalized Thouless pumps for non-invertible SPT phases.

5. Concrete Examples

The general theory is illustrated with two specific classes:

$2\text{Rep}(G_0) \boxtimes 2\text{Vec}_{G_0}$ SPTs: Corresponding to the $G_0$ -cluster state. The authors analyze interfaces between these and trivial phases, showing the symmetry algebra reduces to matrix categories (e.g., $\text{Mat}_{|G_0|}(\text{Vec})$ ) for distinct phases, enforcing degeneracy.
Tambara-Yamagami SPTs: Corresponding to symmetry $2\text{TY}(A, 1)$ . The authors classify these phases and analyze interfaces, recovering known results (e.g., degeneracy of the interface between the ordinary $Z_2$ -cluster model and the trivial phase) and extending them to the non-invertible context.

Significance and Claims

The paper claims to provide the first systematic framework for studying 2+1d SPT phases with non-invertible symmetries using tensor networks.

Unification: It unifies the description of ground states, symmetry operators, and interface physics under the language of group-theoretical fusion 2-categories and their module categories.
Bulk-Boundary Correspondence: It establishes a robust bulk-boundary correspondence for non-invertible symmetries, proving that distinct SPT phases must host degenerate interfaces. This is a stronger constraint than standard LSM-type constraints, as it applies even to gapless interfaces.
Topological Pumping: It demonstrates that topological pumping phenomena (Thouless pumps) persist and are well-defined for non-invertible symmetries, pumping modes that are classified by the representation theory of the strip 2-algebra.
Mathematical-Physical Bridge: The work connects the mathematical classification of fiber 2-functors (from [90]) with explicit lattice constructions, offering a concrete realization of abstract categorical classifications of SPT phases.

The authors emphasize that their construction relies on the assumption that the relevant fusion 2-categories are group-theoretical (admitting a fiber 2-functor), which is necessary for the existence of SPT phases in 2+1d. The results are presented as exact for the constructed lattice models, providing a microscopic foundation for the categorical descriptions of non-invertible symmetries.