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arxiv: 2509.22051 · v2 · submitted 2025-09-26 · ❄️ cond-mat.str-el · hep-th· math-ph· math.MP

From gauging to duality in one-dimensional quantum lattice models

Bram Vancraeynest-De Cuiper , Jos\'e Garre-Rubio , Frank Verstraete , Kevin Vervoort , Dominic J. Williamson , Laurens Lootens This is my paper

Pith reviewed 2026-05-18 13:09 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thmath-phmath.MP
keywords gaugingdualitymatrix product operatorsone-dimensional modelsquantum symmetriescategorical symmetriesbackground fields
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The pith

Gauging and duality transformations are equivalent up to constant depth quantum circuits in one-dimensional quantum lattice models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that gauging and duality transformations are equivalent up to constant depth quantum circuits in one-dimensional quantum lattice models. This equivalence is shown using matrix product operators that act as the lattice representation theory for global categorical symmetries and classify duality transformations. A sympathetic reader would care because these tools are central to understanding symmetries in many-body physics, and their equivalence can simplify analysis of quantum phases and allow clearer handling of background fields. The construction makes the symmetries of the gauged theory manifest.

Core claim

Gauging and duality transformations are shown to be equivalent up to constant depth quantum circuits in the case of one-dimensional quantum lattice models. This is demonstrated by making use of matrix product operators, which provide the lattice representation theory for global (categorical) symmetries as well as a classification of duality transformations. The construction makes the symmetries of the gauged theory manifest and clarifies how to deal with static background fields when gauging generalised symmetries.

What carries the argument

Matrix product operators providing the lattice representation theory for global categorical symmetries and a classification of duality transformations.

If this is right

  • The symmetries of the gauged theory are made manifest.
  • Static background fields can be handled when gauging generalised symmetries.
  • Gauging and duality can be interchanged using only constant depth quantum circuits.
  • Duality transformations are classified within the same framework as symmetries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This equivalence may allow for easier construction of dual models in 1D by using gauging procedures.
  • It could be tested in specific models like spin chains to confirm the constant depth.
  • Connections to higher-dimensional models might reveal where this equivalence breaks down.

Load-bearing premise

Matrix product operators provide the lattice representation theory for global (categorical) symmetries as well as a classification of duality transformations.

What would settle it

A calculation in a concrete 1D model such as the Ising chain showing that the gauged and dual versions require a circuit depth greater than constant to be related.

read the original abstract

Gauging and duality transformations, two of the most useful tools in many-body physics, are shown to be equivalent up to constant depth quantum circuits in the case of one-dimensional quantum lattice models. This is demonstrated by making use of matrix product operators, which provide the lattice representation theory for global (categorical) symmetries as well as a classification of duality transformations. Our construction makes the symmetries of the gauged theory manifest and clarifies how to deal with static background fields when gauging generalised symmetries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that gauging and duality transformations are equivalent up to constant-depth quantum circuits for one-dimensional quantum lattice models. This equivalence is demonstrated by employing matrix product operators (MPOs), which serve as the lattice representation theory for global categorical symmetries and provide a classification of duality transformations. The construction renders the symmetries of the gauged theory manifest and clarifies the treatment of static background fields for generalized symmetries.

Significance. If the central claim is established rigorously, the result unifies two foundational tools in many-body physics at the lattice level for 1D systems with generalized symmetries. The MPO-based approach supplies a concrete representation-theoretic foundation that could streamline constructions of symmetry-enriched phases and duality maps, with potential implications for quantum simulation and topological order. The explicit handling of background fields is a useful practical contribution.

major comments (1)
  1. The central claim equates gauging with duality via constant-depth circuits by invoking MPOs as both the lattice representation theory for categorical symmetries and the classifier of dualities. This step is load-bearing because an MPO with fixed bond dimension still requires an explicit decomposition into a depth-O(1) quantum circuit (independent of chain length L) whose gates implement the duality map while preserving the gauged theory's symmetries. If the construction only shows an abstract MPO correspondence without proving that the circuit depth remains bounded when the symmetry category is non-invertible or when static background fields are included, the equivalence fails to hold beyond invertible cases.
minor comments (2)
  1. Notation for the MPO bond dimension and its relation to the symmetry category should be introduced earlier and used consistently throughout the derivations.
  2. A brief comparison table or explicit example contrasting the new construction with prior invertible-symmetry results would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the potential unifying implications of the MPO-based approach. We address the major comment below and will incorporate clarifications to strengthen the presentation of the circuit decomposition.

read point-by-point responses

  1. Referee: The central claim equates gauging with duality via constant-depth circuits by invoking MPOs as both the lattice representation theory for categorical symmetries and the classifier of dualities. This step is load-bearing because an MPO with fixed bond dimension still requires an explicit decomposition into a depth-O(1) quantum circuit (independent of chain length L) whose gates implement the duality map while preserving the gauged theory's symmetries. If the construction only shows an abstract MPO correspondence without proving that the circuit depth remains bounded when the symmetry category is non-invertible or when static background fields are included, the equivalence fails to hold beyond invertible cases.

    Authors: We thank the referee for this precise and load-bearing observation. In the manuscript the duality map is realized by an MPO whose bond dimension is fixed solely by the data of the symmetry category (or fusion category) and is therefore independent of system size L. In one dimension any such fixed-bond-dimension MPO admits an exact decomposition into a quantum circuit of depth O(1) (with the constant set by the bond dimension and the local Hilbert-space dimension). The decomposition proceeds by sequentially applying local unitary gates that realize the MPO tensors; because the number of layers needed to contract or implement the MPO is bounded by the bond dimension, the circuit depth remains independent of L. The same construction applies verbatim to non-invertible symmetries, since the categorical structure is already encoded in the fixed bond dimension and the fusion rules of the MPO. Static background fields are incorporated as fixed, finite-support defects or insertions into the MPO tensors; these defects do not increase the bond dimension and can be absorbed into the same constant-depth circuit without additional layers. We will add a new subsection (and an accompanying figure) that explicitly carries out the MPO-to-circuit conversion for both an invertible and a non-invertible example, including the case with a non-trivial background field, thereby making the depth bound fully rigorous. revision: yes

Circularity Check

0 steps flagged

Minor self-citation risk from established MPO framework; central equivalence claim remains independent

full rationale

The derivation uses matrix product operators to represent categorical symmetries and classify dualities, then constructs an explicit equivalence to gauging via constant-depth circuits while making symmetries manifest and handling background fields. This builds on prior tensor-network literature (including work by overlapping authors) but does not reduce the main result to a self-referential definition, a fitted parameter renamed as prediction, or an unverified uniqueness theorem. The construction supplies new content by linking gauging directly to duality maps in 1D lattices, remaining self-contained against external benchmarks in MPO theory and quantum circuit depth analysis. No load-bearing step collapses to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that matrix product operators furnish a complete representation theory and duality classification for 1D lattice symmetries; no explicit free parameters or new entities are identified from the abstract.

axioms (1)
  • domain assumption Matrix product operators provide the lattice representation theory for global (categorical) symmetries as well as a classification of duality transformations.
    Invoked in the abstract as the key technical tool enabling the equivalence proof.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Continuum limit of gauged tensor network states

    hep-th 2025-11 unverdicted novelty 6.0

    The continuum limit of gauged tensor networks is well defined and produces a new class of states for non-perturbative continuum gauge theories.

Reference graph

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