From gauging to duality in one-dimensional quantum lattice models

This paper demonstrates that gauging and duality transformations in one-dimensional quantum lattice models are equivalent up to constant depth quantum circuits by utilizing matrix product operators to represent global symmetries and classify duality transformations, thereby clarifying the handling of static background fields in generalized symmetry gauging.

The Gist

Imagine you are a master chef trying to understand the secret recipes of a complex kitchen. In the world of quantum physics, this "kitchen" is a line of atoms (a 1D lattice), and the "recipes" are the rules that govern how these atoms interact.

For a long time, physicists have had two powerful tools to understand these recipes: Gauging and Duality.

• Gauging is like taking a secret ingredient (a symmetry) and making it a local rule that every atom must follow individually. It's like turning a "family rule" (everyone must eat dinner at 6 PM) into a "local law" (every person must check their watch and eat exactly at 6 PM, no matter where they are).
• Duality is like translating a recipe from French to Italian. The ingredients might look different, and the instructions might change, but the final dish (the physics) tastes exactly the same.

This paper, written by a team of physicists, reveals a stunning secret: In one-dimensional quantum systems, these two tools are actually the same thing. They are just different ways of looking at the same transformation.

Here is how they explain it using simple analogies:

1. The "Symmetry" as a Dance

Imagine the atoms in your quantum chain are dancers.

• Global Symmetry: They all dance the same move at the same time. If you rotate the whole stage, the dance looks the same.
• The Problem: Sometimes, the rules of the dance are so complex (involving "non-invertible" symmetries, which are like dances that can't be simply undone) that standard math tools fail.

2. The "Algebra Object" (The Magic Blueprint)

The authors introduce a concept called an Algebra Object. Think of this as a Magic Blueprint or a Master Key.

• In the past, if you wanted to "gauge" a symmetry (make it a local rule), you needed a specific type of blueprint.
• This paper shows that if you pick a specific blueprint (an algebra object), you can build a machine that turns your "Global Dance" into a "Local Dance."

3. The "Gauging Machine" vs. The "Duality Translator"

The paper builds a specific machine (a mathematical circuit) that does the gauging.

• The Old View: You think you are building a "Gauging Machine" that forces every atom to obey the local rule.
• The New View: The authors prove that this exact same machine is actually a Duality Translator. It doesn't just force rules; it translates the entire system into a new language where the rules look different but the physics is identical.

The Analogy:
Imagine you have a book written in a secret code (the original quantum model).

• Gauging is like taking that book and rewriting every sentence so that every word is defined by its neighbors.
• Duality is like translating that book into a different language.
• The Paper's Discovery: They found that the process of rewriting the sentences (Gauging) is mathematically identical to the process of translating the language (Duality). You aren't doing two different things; you are doing the same thing with a different label.

4. The "Quantum Circuit" (The Shortcut)

Usually, translating a quantum system is hard and requires a long, complicated process. However, the authors show that this transformation can be done with a Constant-Depth Quantum Circuit.

The Metaphor:
Imagine you have a tangled ball of yarn (the complex quantum state).

• Usually, untangling it requires pulling one thread at a time, which takes forever (a long circuit).
• This paper shows that you can use a specific "magic knot" (the constant-depth circuit) to untangle the whole ball in a single, quick motion. It's like having a shortcut that instantly rearranges the yarn without getting stuck.

5. Why Does This Matter?

This discovery is a big deal for three reasons:

1. It Unifies the Field: It connects two major areas of physics (Gauging and Duality) that were previously thought to be separate. It's like realizing that "baking a cake" and "making a soufflé" use the exact same oven, just with different settings.
2. It Handles the "Weird" Stuff: It works even when the symmetries are "non-invertible" (the weird, complex dances that can't be undone). This opens the door to studying exotic materials that were previously too hard to understand.
3. It Helps Build New Computers: By understanding how to translate these systems easily, scientists can design better quantum computers and new materials with specific properties (like "short-range entangled states," which are like perfectly synchronized dancers that don't need to hold hands to stay in step).

Summary

In short, this paper says: "If you want to turn a global rule into a local rule in a 1D quantum chain, you are actually just translating the system into a dual language. And the best part? You can do this translation instantly with a simple, short-cut circuit."

It turns a complex, abstract mathematical problem into a clear, unified picture, showing that the "Gauging" and "Duality" tools are two sides of the same coin.

Technical Summary

1. Problem Statement

The paper addresses the theoretical relationship between two fundamental concepts in many-body physics: gauging (promoting a global symmetry to a local gauge symmetry) and duality transformations (mapping a theory to a dual theory with different variables but equivalent physics).

While the equivalence between gauging and duality is well-understood for invertible symmetries (described by groups) in one dimension, the situation is less clear for non-invertible symmetries. These symmetries are described by fusion categories rather than groups. The authors aim to:

• Explicitly construct a gauging map for fusion-categorical symmetries in 1D lattice models.
• Demonstrate that this gauging map is unitarily equivalent to a duality operator (specifically, a Matrix Product Operator or MPO) via a constant-depth quantum circuit.
• Clarify the role of algebra objects in encoding the choice of gauging and how this relates to the Morita equivalence of fusion categories.

2. Methodology

The authors utilize the framework of Tensor Networks, specifically Matrix Product Operators (MPOs), to represent symmetries and dualities on the lattice.

• Fusion Categories and MPOs: They model global symmetries using a unitary fusion category $\mathcal{C}$. The symmetry operators are represented as MPOs, where the virtual indices correspond to objects in $\mathcal{C}$ and the physical indices correspond to a module category $\mathcal{R}$ over $\mathcal{C}$.
• Algebra Objects for Gauging: To gauge a subsystem of the symmetry, they select a haploid symmetric separable Frobenius algebra object $A$ within the fusion category $\mathcal{C}$. This object encodes:
  • The subset of symmetries to be gauged.
  • A choice of generalized discrete torsion.
• Construction of the Gauging Map:
  1. They define a local Gauss law projector $P$ using the multiplication ($\mu$) and comultiplication ($\Delta$) maps of the algebra object $A$.
  2. The global gauging map $G_A$ is constructed by applying these projectors to a trivial gauge field configuration (defined by the unit of the algebra) and projecting onto the gauge-invariant subspace.
  3. This map acts on the original Hilbert space, effectively "gauging" the symmetry.
• Duality MPOs: They review the construction of duality operators from previous work (Lootens et al.), which are MPOs that intertwine local operator algebras of two different lattice realizations (different module categories $\mathcal{R}$ and $\mathcal{R}'$). These duality operators are labeled by module functors in the category $\text{Fun}_{\mathcal{D}}(\mathcal{R}, \mathcal{R}')$, where $\mathcal{D}$ is the Morita dual of $\mathcal{C}$.

3. Key Contributions

A. Equivalence of Gauging and Duality

The central result is the proof that the gauging map $G_A$ and the duality MPO $D_{X_A}$ are equivalent up to a constant-depth unitary quantum circuit.

• The Circuit: The authors explicitly construct a sequence of local unitary gates that transforms the gauging map into the duality operator.
• Mechanism: The proof relies on the internal hom reconstruction. The algebra object $A$ is identified as the internal hom $\text{Hom}(X_A, X_A)$ for a specific simple module object $X_A$ in the category of duality operators.
• Inflation Trick: The authors use an "inflation" technique to rewrite the comultiplication tensor of the algebra $A$ in terms of the module associators of the duality category. This allows the gauging map to be decomposed into a duality MPO (which implements the change of module category) and local unitary transformations (which remove auxiliary "cup" structures).

B. Symmetry of the Gauged Theory

The paper clarifies the global symmetry of the resulting gauged theory:

• If the original symmetry is $\mathcal{C}$ and the gauging is defined by algebra $A$ (corresponding to module category $\mathcal{M} \cong \text{Mod}_{\mathcal{C}}(A)$), the global symmetry of the gauged model is the Morita dual $\mathcal{C}^*_{\mathcal{M}}$.
• This is shown to be equivalent to the category of $A$-$A$ bimodules, $\text{Bimod}_{\mathcal{C}}(A)$.
• The construction makes the emergence of this new symmetry manifest through the duality operator formalism.

C. Handling Boundary Conditions and Twists

The authors extend their formalism to symmetry-twisted boundary conditions:

• They define gauging maps on Hilbert spaces with topological defects (twists) inserted.
• They demonstrate that the gauging map in the presence of twists maps to a linear combination of duality tubes (MPOs with boundary conditions).
• This confirms that the duality permutes the topological sectors (superselection sectors) of the theory, consistent with the structure of the Drinfel'd center $Z(\mathcal{C})$.

D. Anomalies and Short-Range Entanglement

The paper discusses the conditions for full gaugeability:

• A symmetry is fully gaugeable if the algebra object $A$ corresponds to the regular representation (e.g., $A = \text{Hom}(\mathbb{1}, \mathbb{1})$ in the category of vector spaces).
• They argue that if a symmetry can be completely gauged, there must exist a short-range entangled (SRE) symmetric state.
• Conversely, the existence of an 't Hooft anomaly (obstruction to gauging) implies the absence of such an SRE state. This provides a lattice-based criterion for detecting anomalies in non-invertible symmetries.

4. Results and Examples

The authors validate their formalism with two specific examples:

1. $\text{Rep}(S_3)$:

   • They analyze the fusion category of representations of the symmetric group $S_3$.
   • They identify algebra objects corresponding to subgroups ($Z_2, Z_3$) and the full group.
   • They show how gauging the $Z_2$ or $Z_3$ sub-symmetries leads to specific dualities, recovering known results (like Kramers-Wannier duality) as special cases of this categorical framework.

2. Haagerup Categories ($H_1, H_2, H_3$):

   • They apply the formalism to the Haagerup fusion categories, which are non-group-theoretical and have no known classical group realization.
   • They demonstrate how the Morita equivalence between $H_1, H_2, H_3$ is realized via gauging specific algebra objects (e.g., gauging the $\text{Vec}_{\mathbb{Z}_3}$ sub-symmetry in $H_2$ yields $H_3$).
   • This provides a concrete lattice construction for the dualities between these exotic quantum models.

5. Significance

• Unification: The paper unifies the concepts of gauging and duality in 1D, showing they are two sides of the same coin mediated by constant-depth circuits. This bridges the gap between algebraic approaches (Frobenius algebras) and tensor network approaches (MPOs).
• Non-Invertible Symmetries: It provides the first systematic, explicit construction of gauging maps for non-invertible symmetries on the lattice, moving beyond the limitations of group-based gauging.
• Computational Utility: By expressing gauging as a constant-depth circuit equivalent to a duality MPO, the work offers practical tools for simulating gauged theories using tensor network methods (e.g., DMRG, PEPS) without needing to explicitly introduce gauge fields in the simulation.
• Anomaly Detection: The link between full gaugeability and the existence of SRE states offers a new diagnostic tool for identifying anomalies in gapped phases protected by non-invertible symmetries.
• Generalization: The framework is set up to be generalizable to higher dimensions and potentially to spacetime symmetries (though the latter requires further work on orientation-reversing domain walls).

In summary, this work establishes a rigorous, constructive dictionary between the algebraic data of fusion categories (algebra objects) and the physical operations of gauging and duality in quantum lattice models, significantly advancing the understanding of non-invertible symmetries in condensed matter physics.