🔬 condensed matter

Symmetry-permuting entanglers for non-invertible symmetry-protected topological phases

This paper demonstrates that distinct 1+1D symmetry-protected topological phases with non-invertible $\mathrm{Rep}(D_8)$ symmetry, which cannot be connected by standard symmetric entanglers, can instead be linked via symmetry-permuting finite-depth local unitary circuits that realize TQFT-predicted dualities and reproduce anyon permutations in string order parameters.

Original authors: Minyoung You

Published 2026-09-29

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CC BY 4.0

Original authors: Minyoung You

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 subatomic world, matter does not always behave like solid objects; sometimes, it behaves like a pattern of connections. Physicists study these patterns to understand a special kind of order called a topological phase. Imagine a knot in a string: you cannot untie it without cutting the string, no matter how you twist or turn it. Similarly, certain materials possess a hidden structure that cannot be changed by gentle, local adjustments. These are the topological phases. For decades, scientists have focused on materials protected by "invertible" symmetries, which are like mirror reflections or rotations where every action has a clear, reversible opposite. In these familiar cases, researchers know how to build a bridge between different phases using a specific type of quantum circuit, a mathematical tool that rearranges the connections between particles without breaking the underlying symmetry.

However, the universe also contains stranger symmetries, known as non-invertible symmetries. These are like a rule where you can combine two actions to get a third, but you cannot simply reverse the process to get back to the start. For a long time, it was believed that materials protected by these strange, non-invertible symmetries were fundamentally disconnected. The prevailing thought was that if you had two different versions of such a material, there was no way to transform one into the other without destroying the symmetry entirely. It seemed as though these phases were isolated islands, separated by an unbridgeable gap. This belief created a puzzle: if the laws of physics allow for these different phases, how do they relate to one another?

A researcher has now solved this puzzle by discovering a new kind of bridge. They focused on a specific, complex material model that possesses a non-invertible symmetry related to the representation category of the dihedral group D8. Previous work had shown that three distinct versions of this material could not be connected by any standard, symmetry-preserving bridge. The researcher proved that while a direct, static bridge was impossible, a dynamic bridge existed that could shuffle the rules of the symmetry itself. They constructed a precise, finite-depth quantum circuit—a sequence of local operations that acts locally and with limited depth—that transforms one of these material phases into another. Crucially, this circuit does not keep the symmetry rules fixed; instead, it permutes them, swapping one type of symmetry rule for another while preserving the overall structure of the system.

The researcher did not just guess at this solution; they built it from the ground up using a method based on how these symmetry rules interact locally. They started with the mathematical description of the material's ground state and worked backward to find the specific sequence of operations that would turn a simple, unentangled state into the complex, topological state. They found that the circuit acts like a translator that changes the language of the symmetry. When they applied this circuit to the material, it successfully converted the first phase into a second, distinct phase. They then verified that this transformation was not random by tracking how the circuit affected the "string order parameters," which are the physical signatures of the material's hidden topological order. The circuit rearranged these signatures in a way that perfectly matched the predictions of a theoretical framework called topological quantum field theory.

This discovery changes our understanding of how these exotic materials relate to one another. The researcher showed that the three distinct phases of this D8-symmetric material are not isolated. Instead, they are connected by a group of transformations that act like a permutation of the symmetry rules themselves. By combining their new circuit with a simpler, known transformation, they demonstrated that all three phases can be reached from one another. This confirms a specific prediction from theoretical physics: that while you cannot connect these phases with a circuit that leaves the symmetry rules untouched, you can connect them with a circuit that shuffles those rules. The work provides a concrete, microscopic realization of a concept that was previously only understood as an abstract mathematical possibility.

The implications of this finding extend beyond this single material model. It suggests that the landscape of quantum phases is more interconnected than previously thought. The researcher established that the obstruction to connecting these phases is not absolute; it simply requires a different kind of tool. By developing a method to construct these "symmetry-permuting" circuits, they have opened a new path for exploring non-invertible symmetries. Their work proves that even in systems where the usual rules of symmetry breaking do not apply, there are still deep, structural connections waiting to be discovered. The ability to map one phase to another by rearranging the symmetry rules offers a new way to classify and understand the rich variety of quantum matter that exists in nature.

Technical Summary: Symmetry-Permuting Entanglers for Non-Invertible SPT Phases

Problem Statement
One-dimensional symmetry-protected topological (SPT) phases with invertible symmetries are typically characterized by symmetric entanglers: finite-depth local unitary (FDLU) circuits that generate representative ground states from a product state while commuting with the global symmetry. However, for systems with non-invertible symmetries (described by fusion categories), this correspondence breaks down. Specifically, distinct SPT phases protected by the $\text{Rep}(D_8)$ symmetry cannot be connected by any symmetric entangler (a circuit commuting with all symmetry operators). While previous work established the existence of "fixed-charge dualities" (FCDs) for invertible symmetries, the classification of non-invertible SPT phases involves "fixed-algebra dualities" (FADs), where the symmetry boundary is preserved as a condensable algebra, but its simple summands (symmetry lines) may be permuted. The central problem addressed is whether these distinct $\text{Rep}(D_8)$ SPT phases, which are disconnected by symmetric entanglers, can be connected by a finite-depth circuit that permutes the symmetry lines, thereby realizing the FADs predicted by the symmetry topological quantum field theory (symTFT).

Methodology
The author constructs an explicit microscopic realization of such a connection using Matrix Product Unitaries (MPUs). The methodology proceeds as follows:

Theoretical Framework: The work utilizes the symTFT description where dualities are encoded by braided autoequivalences of the bulk Drinfeld center $Z(\mathcal{C})$ . The author focuses on FADs, which preserve the Lagrangian algebra of the symmetry boundary but permute its simple summands.
Local Constraints: Instead of searching for a global circuit ad hoc, the author derives local constraints for the MPU tensor $M$ . They start with the global symmetry-permuting condition $U S_a U^\dagger = S_{\phi(a)}$ , where $S_a$ are symmetry MPOs and $\phi$ is a permutation of simple objects.
Crossing Relations: By vectorizing the MPOs and applying the fundamental theorem of matrix product states, they derive "pulling-through" equations. These equations relate the MPU tensor $M$ , the symmetry MPO tensor $T_{S_a}$ , and a "crossing tensor" $X_a$ that intertwines the virtual spaces.
Ansatz and Solution: For the $\text{Rep}(D_8)$ case, the author targets a specific FAD that exchanges the trivial SPT phase with an SPT phase associated with a subgroup $H = \langle r^2, s \rangle$ . They impose a bond-dimension-two ansatz for the MPU. The crossing tensors $X_a$ are determined by the requirement that the MPU maps the action tensors of the product state to those of the target SPT state.
Construction: Solving the resulting linear constraints (crossing relations) and imposing unitarity, hermiticity, and the exact mapping of fixed-point states yields an explicit MPU tensor. The resulting circuit is shown to be an FDLU (finite-depth local unitary) with a trivial index.

Key Contributions and Results

Explicit Construction: The paper constructs an explicit bond-dimension-two MPU that connects the trivial product state to a non-trivial $\text{Rep}(D_8)$ SPT phase. This circuit does not commute with the symmetry MPOs individually but permutes them according to the target FAD.
Symmetry Permutation: The constructed circuit realizes the specific permutation of symmetry lines: $\rho_a \leftrightarrow \rho_c$ , while $\rho_b$ and $\sigma$ are fixed. This confirms that distinct SPT phases can be connected via symmetry-permuting entanglers, resolving the apparent obstruction found for symmetric entanglers.
Microscopic Anyon Permutation: The author demonstrates that the action of this MPU on string order parameters (which represent anyons in the symTFT) reproduces the anyon permutation predicted by the TQFT duality. Specifically:
- Invertible strings transform as $([e], \rho_a) \to ([r^2], \rho_c)$ and $([e], \rho_c) \to ([r^2], \rho_a)$ .
- The non-invertible string $\sigma$ transforms such that the endpoint charges are permuted, mapping $([e], \sigma) \to ([s], \omega_2)$ .
Generation of the Full Duality Group: By combining this non-on-site symmetry-permuting entangler with an on-site unitary circuit (induced by a group automorphism of $D_8$ ), the author shows that the full $S_3$ group of FADs is generated, connecting all three distinct $\text{Rep}(D_8)$ SPT phases.

Significance
The paper provides a concrete microscopic resolution to the classification of non-invertible SPT phases. It substantiates the symTFT prediction that while symmetric entanglers (FCDs) may not exist between certain non-invertible SPT phases, finite-depth circuits realizing fixed-algebra dualities (FADs) do exist. This work moves beyond the "ad hoc" construction of previous examples (such as for $\text{Rep}(A_4)$ ) by deriving the circuit from local crossing relations and action tensors. The results strengthen the proposed microscopic interpretation of the symTFT classification, showing that the distinction between fixed-charge and fixed-algebra dualities has a direct realization in quantum circuits. The author notes that while the construction is guided by local relations, it is not yet fully algorithmic, and the extraction of anyon permutations from string operators does not uniquely determine the underlying braided autoequivalence in all cases.

Technical Summary: Symmetry-Permuting Entanglers for Non-Invertible SPT Phases

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