🔬 condensed matter

Disentangling the Toric Code

This paper demonstrates that while the non-on-site $U(1)$ symmetries of commuting-projector Hamiltonians like the 1+1d Ising model and 2+1d toric code cannot be made on-site using finite-dimensional ancillae due to fractionalized excitations, they can be disentangled into on-site symmetries by employing infinite-dimensional ancillae, provided the resulting ancilla Hamiltonians are unbounded from below.

Original authors: Lei Gioia, Salvatore D. Pace, Ruben Verresen, Shu-Heng Shao, Ryan Thorngren

Published 2026-09-28

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CC BY 4.0

Original authors: Lei Gioia, Salvatore D. Pace, Ruben Verresen, Shu-Heng Shao, Ryan Thorngren

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 microscopic world of quantum materials, physicists often look for hidden rules that govern how particles behave together. One such rule is symmetry, a property where a system looks the same after being rotated or shifted, much like a snowflake looks identical after a turn. In the smooth, continuous world of theoretical physics, these symmetries are well understood and rarely cause trouble. However, when scientists try to build these systems on a computer or a grid of atoms, a strange problem arises. The rigid structure of the grid can sometimes clash with the symmetry, creating an "anomaly" that prevents the system from behaving as expected. This clash is not just a mathematical quirk; it dictates what kinds of matter can exist and how they can be manipulated. Understanding whether these symmetries can be made to work smoothly on a grid is crucial for building future quantum computers and for understanding the fundamental laws of nature.

A team of researchers recently tackled a specific version of this problem involving two famous models of quantum matter: a simple magnetic chain and a complex grid known as the toric code. These models are special because their energy levels are perfectly spaced, like rungs on a ladder, and they host exotic particles that cannot be found in ordinary matter. The researchers asked a fundamental question: can the symmetry governing these systems be "untangled" so that it acts locally on each atom, rather than requiring a complicated, global connection across the entire grid? The answer depends entirely on what tools are allowed to help. If the scientists are restricted to using only standard, finite-sized quantum bits, the answer is a definitive no. The exotic particles in these systems, such as domain walls and anyons, create an obstruction that cannot be removed. The symmetry remains stubbornly non-local, and the system retains a deep, long-range order that resists simplification.

However, the story changes dramatically when the researchers allow for a different kind of helper: an infinite-dimensional system, which can be thought of as a continuous dial rather than a simple switch. By introducing these unbounded helpers, the team proved that both the magnetic chain and the toric code can be completely disentangled. The complex, global symmetry can be transformed into a simple, local one that acts independently on each site. This result is surprising because it shows that the obstruction to simplifying these systems is not an absolute law of nature, but rather a limitation of the tools used to study them. When the right kind of infinite helper is added, the systems become trivial, revealing that the "anomaly" was an artifact of the finite grid, not a fundamental barrier.

The researchers demonstrated this by constructing explicit mathematical circuits that act as a disentanglement machine. For the magnetic chain, they showed that the symmetry could be broken down into local pieces if they added a continuous variable at every point. For the more complex toric code, which involves particles that braid around each other in two dimensions, the task was harder. They had to introduce not only continuous variables but also specific types of fermionic particles to successfully untangle the system. Once these helpers were in place, the researchers could transform the complex Hamiltonian, which describes the energy of the system, into a simple sum of independent parts. This means the ground state, which usually holds deep, long-range entanglement, could be converted into a simple, short-range state.

This work clarifies a long-standing tension between how physicists view matter on a grid versus in a continuous field theory. In the continuous world, there are no anomalies for these specific symmetries in two dimensions, meaning they should be able to exist without obstruction. The grid, however, seemed to forbid this. The paper proves that the grid's obstruction is real and insurmountable if one is limited to finite resources, but it vanishes completely when infinite resources are permitted. This suggests that the grid and the continuous world are not fundamentally at odds; they simply require different sets of tools to reveal their true nature. The findings confirm that while certain exotic states of matter cannot be simplified on a finite grid, they are not inherently anomalous. They are merely waiting for the right kind of infinite-dimensional assistance to reveal their underlying simplicity.

The implications of this discovery extend beyond these two specific models. The researchers argue that their methods likely apply to a wide class of topological systems, including those used to describe quantum double models and string-net condensates. The key takeaway is that the "non-invertible" nature of these obstructions—meaning they cannot be fixed by simply adding another finite system—is a feature of finite-dimensional physics. Once the dimension is allowed to grow without bound, the obstructions disappear, and the symmetries align perfectly with the predictions of continuous quantum field theory. This provides a bridge between the discrete world of lattice models and the smooth world of field theory, showing that they describe the same reality, provided one is willing to use the full power of infinite-dimensional mathematics to unlock it.

Technical Summary: Disentangling the Toric Code

Problem Statement
The paper investigates the "on-siteability" of global $U(1)$ symmetries generated by commuting-projector Hamiltonians with integer spectra. Specifically, it addresses whether such symmetries, which are inherently non-on-site (acting as global unitaries rather than tensor products of local unitaries), can be transformed into on-site symmetries via finite-depth quantum circuits (FDQC) when augmented with ancillary degrees of freedom. The study focuses on two primary examples: the $U(1)$ symmetry generated by the 1+1d Ising ferromagnet ( $H_{SSB}$ ) and the $U(1)$ symmetry generated by the 2+1d Toric Code Hamiltonian ( $H_{TC}$ ).

The central question is how the nature of the ancilla—specifically whether it is semi-bounded (bounded from below/above, including finite-dimensional systems) or unbounded (infinite-dimensional, such as rotors)—affects the ability to "disentangle" these symmetries. This inquiry is motivated by the broader goal of reconciling lattice definitions of anomalies with continuum quantum field theory (QFT) predictions, particularly regarding whether obstructions to on-siteability correspond to non-trivial Symmetry Protected Topological (SPT) phases.

Methodology
The authors employ a combination of algebraic and constructive techniques:

Ground State Algebra Analysis: For semi-bounded ancillae, the authors utilize the concept of a "ground state algebra" ( $A_{gs}$ ). They define a ground state algebra as the algebra of operators preserving the ground state subspace. A key lemma establishes that if a Hamiltonian is on-siteable via a finite-depth circuit (or spread- $s$ Quantum Cellular Automaton, QCA), its ground state algebra must admit a specific factorization property over spatial bipartitions.
Obstruction via Non-Factorizability: The authors demonstrate that for systems with fractionalized excitations (domain walls in the Ising model, anyons in the Toric Code), certain global symmetry operators (e.g., $\prod X_j$ ) cannot be factorized within the ground state algebra without creating excitations at the boundaries of the partition. This non-factorizability serves as an obstruction to on-siteability.
Constructive Disentanglement: For unbounded ancillae (specifically rotors with conjugate number operators), the authors employ explicit construction of disentangling circuits. They utilize the concept of "local factorizability," showing that if a symmetry generator can be written as a sum of mutually commuting local terms (possibly with boundary terms), it can be made on-site using rotors.
Spectral Analysis: The paper analyzes the energy spectra of the Hamiltonians, noting that while the original Hamiltonians have integer spectra, the disentangled versions involving unbounded ancillae result in unbounded spectra, rendering them unsuitable as physical Hamiltonians for a ground state but valid as symmetry generators.

Key Contributions and Results

Obstruction with Semi-Bounded Ancillae:
- Proposition 3 & 4: The $U(1)_{SSB}$ symmetry (Ising ferromagnet) cannot be made on-site with semi-bounded ancillae. The obstruction arises because the global spin-flip operator $\prod X_j$ is non-factorizable in the ground state algebra. The required circuit depth scales with system size ( $D \sim L$ ), preventing finite-depth on-siteability.
- Proposition 6 & Corollary 1: Similarly, the $U(1)_{TC}$ symmetry (Toric Code) is not on-siteable with semi-bounded ancillae. The lattice 1-form symmetry operators (string operators creating anyon pairs) are non-factorizable due to their mutual braiding statistics. This obstruction is "non-invertible," meaning no other finite-dimensional Hamiltonian can be added to cancel the anomaly.
- Hall Conductance Argument: An alternative proof for the Toric Code obstruction uses the Hall conductance. The $U(1)_{TC}$ symmetry exhibits a non-zero Hall conductance ( $\sigma_H = 1 \mod 2$ ), which is impossible for an on-site symmetry in a commuting-projector Hamiltonian with finite-dimensional local Hilbert spaces.
Disentanglement with Unbounded Ancillae:
- Proposition 5: The $U(1)_{SSB}$ symmetry becomes on-siteable in finite depth when infinite-dimensional unbounded ancillae (rotors) are introduced. The global symmetry is locally factorized into commuting $U(1)$ representations, allowing a finite-depth circuit to map the Hamiltonian to a sum of local rotor number operators.
- Propositions 8, 10, & 12: The $U(1)_{TC}$ $U (1)_{T C}$ symmetry (and the enhanced $U(1)_e \times U(1)_m$ $U (1)_{e} \times U (1)_{m}$ ) is also on-siteable with unbounded ancillae. The authors provide explicit constructions:
  - Using integer-charged rotors and fermionic ancillae (Prop 8).
  - Using purely integer-charged rotors (Prop 10), constructing a complex circuit involving gauge-invariant combinations of rotor angles and lattice operators to disentangle the Hamiltonian.
  - A simplified model ( $H'_{TC}$ ) with integer anyon charges is shown to be on-siteable even more straightforwardly (Prop 12).
- Short-Range Entangled (SRE) States: The paper demonstrates that while the ground states of these systems are Long-Range Entangled (LRE), symmetric SRE eigenstates exist in the presence of unbounded ancillae (Corollary 2).

Significance and Claims
The paper claims that the presence of infinite-dimensional unbounded ancillae resolves the apparent contradiction between lattice obstructions and continuum QFT expectations.

Alignment with QFT: In 2+1d, continuum QFT predicts no anomalies for global $U(1)$ symmetries (equivalently, no non-trivial $U(1)$ SPT phases in 3+1d). The paper shows that with unbounded ancillae, the Toric Code symmetry is on-siteable, aligning the lattice result with the continuum prediction.
Nature of Lattice Anomalies: The results suggest that obstructions to on-siteability with finite-dimensional ancillae are distinct from standard SPT classifications. These obstructions are "non-invertible" and cannot be characterized by standard group cohomology.
Role of Ancillae: The paper argues that semi-bounded ancillae (finite-dimensional) are insufficient to trivialize these symmetries because the obstruction relies on the structure of the ground state subspace, which remains well-defined. Unbounded ancillae, however, alter the spectral properties (making the spectrum unbounded), effectively removing the ground state subspace constraint and allowing for disentanglement.

The authors conclude that while the disentangled Hamiltonians are unbounded and thus not physical ground-state Hamiltonians, they serve as valid generators for compact, locality-preserving $U(1)$ global symmetries, bridging the gap between lattice and continuum anomaly theories.

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