Entropic order parameters and topological holography

This paper demonstrates that the Symmetry Topological Field Theory (SymTFT) framework offers a natural and intuitive approach to defining entropic order parameters for phases with broken symmetries, including non-invertible ones, by revealing the information-theoretic origins of vacuum distinguishability through the exclusion of specific operators from observation.

The Gist

The Big Picture: Mapping the Unseen

Imagine you are trying to understand the different "moods" or "states" of a complex system, like a crowd of people at a concert or the magnetic spins in a piece of metal. In physics, these states are called phases.

For a long time, physicists used a specific tool to tell these phases apart: the Order Parameter. Think of this like a thermometer. If the temperature is high, the water is liquid; if it's low, it's ice. In the old "Landau" way of thinking, if a system breaks its symmetry (like a magnet choosing a specific North/South direction), you look for a specific local signal (like a needle pointing North) to prove it.

The Problem: In very complex, "strongly coupled" systems (where everything is tangled up), finding that specific needle is incredibly hard. Sometimes, the needle doesn't exist at all, or there are too many to count.

The New Tool: This paper introduces a new way to measure these phases using Information Theory. Instead of looking for a single needle, they ask: "How much information do we lose if we ignore the messy, complicated parts of the system?" They call this the Entropic Order Parameter. It's like measuring the "confusion" or "surprise" in the system.

The Magic Sandwich: Topological Holography (SymTFT)

To make this calculation easier, the authors use a clever trick called Symmetry Topological Field Theory (SymTFT), or "Topological Holography."

Imagine the 2D world where the physics happens (like a flat sheet of paper) is the bottom slice of a sandwich.

• The Bottom Slice (Physical Boundary): This is the real world we are studying. It's messy and dynamic.
• The Top Slice (Symmetry Boundary): This is a special, rigid layer that holds the "rules" of the game (the symmetries).
• The Filling (3D Bulk): Between them is a 3D space filled with invisible, magical threads (topological lines).

How it works:
Instead of trying to solve the messy physics on the bottom slice directly, you look at the 3D filling. The "threads" in the filling connect the top and bottom.

• If a thread can attach to the bottom slice, it represents a specific type of operator (a tool you can use to measure the system).
• If a thread cannot attach to the bottom, it represents a "hidden" or "twisted" part of the system.

This setup separates the rules (topology) from the dynamics (the messy physics). It's like studying a game of chess by looking at the rulebook (the top slice) and the board (the bottom slice) separately, rather than trying to predict every move in real-time.

The "Intertwiners": The Messengers

In this framework, there are special objects called Intertwiners.

• Analogy: Imagine a messenger who can walk from the "Rule Layer" down to the "Real World."
• If the messenger is "invisible" (trivial), they represent a standard, boring measurement.
• If the messenger carries a "badge" (a non-trivial thread), they represent a special, symmetry-breaking measurement.

When a symmetry is spontaneously broken (the system picks a specific state), these messengers combine to form the different "vacua" (the ground states of the system).

The Big Discovery: Distinguishable Vacua

Here is the most surprising part of the paper, explained simply:

1. Old Symmetries (Invertible/Group Symmetries):
Think of a standard symmetry like a spinning top. If it breaks, it falls to the left or the right.

• The Result: The "Left" state and the "Right" state are indistinguishable in terms of information loss. If you measure them with the new "Entropic Order Parameter," they both show the exact same amount of "confusion" (Relative Entropy). They are twins.

2. New Symmetries (Non-Invertible/Fusion Symmetries):
Now, imagine a more exotic symmetry, like the "Ising" symmetry found in certain quantum materials. These aren't just simple rotations; they are like complex fusion rules (e.g., "If you mix A and B, you get C, but if you mix C and D, you get nothing").

• The Result: When these exotic symmetries break, the resulting ground states are NOT twins.
• The Analogy: Imagine you have three different colored balls (Red, Blue, and Green). In the old world, if you broke the symmetry, you'd get two identical Red balls. In this new world, you might get one Red ball and one Green ball.
• The Measurement: The "Entropic Order Parameter" detects this difference! It tells you that the "Red" vacuum and the "Green" vacuum lose different amounts of information. They are distinguishable.

Why Does This Happen?

The paper explains that this difference comes down to Quantum Dimensions.

• In the old world, every "piece" of the symmetry has a size of 1.
• In the new world, some pieces are "bigger" (have a quantum dimension greater than 1).
• The "Entropic Order Parameter" is essentially a scale that weighs these pieces. If the pieces have different weights, the resulting states (vacua) will have different "information weights," making them unique and distinguishable.

Summary of the Paper's Claims

1. New Framework: The authors use a "sandwich" model (SymTFT) to visualize and calculate how symmetries break in 1D and 2D systems.
2. New Metric: They use Relative Entropy (a measure of information loss) as a universal "Order Parameter" to detect symmetry breaking.
3. Key Finding for Standard Symmetries: When normal symmetries (like $Z_2$ or $S_3$) break, all resulting ground states look the same to this new metric. They are indistinguishable.
4. Key Finding for Exotic Symmetries: When "non-invertible" symmetries (like Rep($S_3$) or Ising) break, the resulting ground states are distinguishable. Some states are "heavier" or "more complex" than others.
5. The "Why": This distinguishability is directly linked to the mathematical "size" (quantum dimension) of the symmetry components.

In a nutshell: The paper provides a new, intuitive way to see that when the universe breaks "exotic" symmetries, the resulting worlds are not all the same—they have unique fingerprints that can be measured by how much information is hidden from us.

Technical Summary

Technical Summary: Entropic Order Parameters and Topological Holography

Problem Statement
The classification of phases in quantum field theories (QFTs), particularly those with strong couplings, remains a central challenge in physics. The traditional Landau paradigm relies on local order parameters (non-vanishing expectation values of local operators) to signal spontaneous symmetry breaking (SSB). However, identifying the pertinent operator is often ambiguous and difficult in strongly coupled systems. Furthermore, the original Landau paradigm does not naturally accommodate "generalized" symmetries, such as higher-form and non-invertible (fusion category) symmetries, which have recently enriched the landscape of symmetry-breaking phases. While information-theoretic quantities like relative entropy (or entropic order parameters) have been proposed to address the limitations of local operators, a unified framework for computing these quantities across both invertible and non-invertible symmetries, and for understanding the distinguishability of resulting vacua, has been lacking.

Methodology
The authors employ the Symmetry Topological Field Theory (SymTFT), also known as topological holography, as a unifying framework. In this construction, a $d$-dimensional QFT with a global symmetry category $\mathcal{C}$ is represented holographically by a $(d+1)$-dimensional topological field theory (the SymTFT) bounded by two surfaces: a "symmetry boundary" (Dirichlet condition) and a "physical boundary."

• SymTFT Construction: The bulk of the SymTFT is a topological field theory whose line operators form the Drinfeld center $\mathcal{Z}(\mathcal{C})$. The symmetry boundary is fixed to the Dirichlet condition, while the physical boundary condition encodes the specific phase of the QFT. Gapped phases correspond to topological physical boundaries, which are classified by indecomposable module categories over $\mathcal{C}$.
• Intertwiners and Vacua: Local operators in the QFT are represented as triples involving a bulk topological line. Operators invariant under the symmetry correspond to trivial bulk lines. The authors introduce intertwiners (non-locally generated operators) associated with bulk topological lines that can end on both boundaries. In the SSB phase, the vacua are constructed as orthogonal idempotents formed by linear combinations of these intertwiners.
• Entropic Order Parameter: The authors utilize relative entropy as the order parameter. They define a conditional expectation map $\mathcal{E}$ that projects the full operator algebra $\mathcal{F}$ onto the symmetry-invariant subalgebra $\mathcal{O}$ by removing non-invariant (non-trivial bulk line) components. The entropic order parameter is defined as the relative entropy $S(v_k | v_k \circ \mathcal{E})$ between a vacuum state $v_k$ and its symmetrized version. This quantity measures the information loss when non-invariant operators are excluded from observation.

Key Contributions and Results

1. Unified Framework for Invertible and Non-Invertible Symmetries:
   The paper demonstrates that the SymTFT construction naturally accommodates both ordinary (invertible) group symmetries and non-invertible (fusion category) symmetries. In both cases, vacua are expressed as linear combinations of intertwiners, analogous to Cardy states in rational conformal field theories.

2. Distinguishability of Vacua in Non-Invertible SSB:
   A primary result is the demonstration that vacua arising from the spontaneous breaking of non-invertible symmetries can be physically distinguishable, unlike those from ordinary group symmetries.

   • Group Symmetries: For ordinary finite groups $G$, the conditional expectation is trace-preserving. The relative entropy is uniform across all vacua: $S = \log(|G|/|H|)$, where $H$ is the unbroken subgroup. Consequently, the relative Euler terms vanish, and vacua are indistinguishable.
   • Non-Invertible Symmetries: For non-invertible symmetries (e.g., $\text{Rep}(G)$ or the Ising fusion category), the conditional expectation is generally not trace-preserving. The relative entropy depends on the specific vacuum $v_x$:
     $$S(v_x | v_x \circ \mathcal{E}) = \log \left( \frac{\dim(\mathcal{M})}{d_x^2} \right)$$
     where $d_x$ is the quantum dimension of the simple object labeling the vacuum, and $\dim(\mathcal{M})$ is the total quantum dimension of the module category. Because $d_x$ varies for different vacua in non-Abelian or non-group-theoretical cases, the relative entropies differ. This leads to non-vanishing relative Euler terms, rendering the vacua distinguishable.

3. Explicit Calculations:
   The authors explicitly compute these quantities for several examples:

   • $Z_2$ and $Z_2 \times Z_2$: Confirms uniform relative entropy for SSB phases and identifies string order parameters in SPT phases.
   • $S_3$ Symmetry: Illustrates the decomposition of irreps and the calculation of vacua for both complete and partial breaking.
   • $\text{Rep}(S_3)$ Symmetry: Shows that vacua labeled by different irreps of $S_3$ have distinct relative entropies (e.g., $\log 6$ vs. $\log 3/2$), confirming their distinguishability.
   • Ising Symmetry: For the non-group-theoretical Ising fusion category, vacua labeled by the identity ($1$), spin flip ($\eta$), and duality ($N$) lines exhibit different entropic order parameters ($\log 4$ vs. $\log 2$), directly linked to their quantum dimensions ($1$ vs. $\sqrt{2}$).

4. Connection to Relative Euler Terms:
   The paper establishes an algebraic link between the distinguishability of vacua and the relative Euler terms. The difference in relative entropies corresponds to the eigenvalues of the relative modular operator, providing a rigorous operator-algebraic characterization of vacuum distinguishability in non-invertible SSB.

Significance
The paper claims that the SymTFT construction provides a "natural and intuitive framework" for characterizing SSB of generalized symmetries. Its significance lies in:

• Separation of Kinematics and Dynamics: It cleanly separates symmetry properties (kinematics) from dynamical details, allowing for the computation of order parameters based solely on the symmetry category and the choice of boundary condition.
• Information-Theoretic Insight: It offers a clear information-theoretic perspective on why non-invertible symmetry breaking leads to distinguishable vacua: the "information loss" (relative entropy) upon restricting to invariant operators varies depending on the quantum dimensions of the specific vacuum sector.
• Unification: It places invertible and non-invertible symmetry breaking on equal footing, showing that the latter is a natural generalization where the "idempotent" structure of vacua leads to non-uniform entropic signatures.

The authors note that while the focus is on gapped $(1+1)$-dimensional phases, the framework is designed to generalize to gapless phases and higher dimensions, though those specific investigations are reserved for future work.