Strong-to-weak spontaneous breaking of 1-form symmetry and intrinsically mixed topological order

This paper introduces an equivalence relation based on finite Lindbladian evolution and Rényi-2 Markov length to classify "intrinsically mixed" topological phases arising from strong-to-weak spontaneous breaking of 1-form symmetries, demonstrating that disordered ensembles of the toric code exhibit these stable phases with distinct symmetry-breaking patterns.

The Gist

The Big Picture: When Order Gets Messy

Imagine you have a perfectly organized library (a pure quantum state). Every book is in its exact spot, and the librarian knows exactly where everything is. In physics, this represents a "perfect" material with Topological Order—a special kind of order that is robust and hard to break, like a secret code hidden in the arrangement of books.

Now, imagine a chaotic storm hits the library. Books get knocked off shelves, some are lost, and the librarian can no longer be 100% sure where every book is. The library is now in a mixed state (described by a density matrix). It's a mix of "maybe the book is here" and "maybe it's there."

For a long time, physicists thought that if you messed up a library enough, the special "secret code" (topological order) would just vanish, leaving you with a completely random pile of books (a trivial state).

This paper argues that there is a "middle ground." Even when the library is messy, it doesn't just become random. It can settle into a new, strange kind of order that is intrinsically mixed. It's not a perfect library, and it's not a total mess; it's a specific type of organized chaos that couldn't exist in a perfect library.

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Key Concept 1: The Two Types of Rules (Symmetries)

In physics, "symmetry" is like a rule that keeps things consistent. The paper introduces two ways a rule can apply to our messy library:

1. Strong Symmetry (The Strict Librarian): The rule is absolute. If you try to move a book, the system fights back immediately. The rule is "hard-coded" into the state.
2. Weak Symmetry (The Lenient Librarian): The rule is only true on average. If you look at the whole library, the books seem balanced, but if you look at one specific shelf, the rule might be broken.

The Discovery: The authors found a scenario where a rule starts as Strong (strict) but, due to the messiness (disorder), degrades into Weak (lenient). They call this "Strong-to-Weak Spontaneous Symmetry Breaking" (SW-SSB).

Analogy: Imagine a rule that says, "Everyone must wear a red hat."

• Strong: Everyone is wearing a red hat, and if you take one off, the system collapses.
• Weak: On average, 50% of people are wearing red hats, so the "spirit" of the rule exists, but no single person is forced to wear one.
• SW-SSB: The system started with a strict rule, but the chaos of the storm forced it to relax into a state where the rule only holds on average. This specific "relaxed but still ordered" state is what the paper calls Intrinsically Mixed.

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Key Concept 2: The New Magnifying Glass (Rényi-2 Markov Length)

How do we tell the difference between a "truly random mess" and this "intrinsically mixed order"?

The old way of looking at these systems was like using a standard flashlight (called Two-Way Channel Connectivity). It was too dim. It couldn't see the difference between the "intrinsically mixed" library and a completely random pile of books. It said, "They both look messy, so they are the same."

The authors invented a super-magnifying glass called the Rényi-2 Markov Length.

• Think of this as a measure of how far you have to look to see if the library is truly random.
• If the "mess" is just normal noise, the correlation dies out quickly (short Markov length).
• If the library has this special "intrinsically mixed" order, the correlation stretches out in a specific way that the old flashlight missed, but this new magnifying glass catches.

The Result: With this new tool, they proved that the "intrinsically mixed" state is a distinct phase of matter. It is not just a broken version of a perfect state; it is a unique state that only exists because of the mix of order and disorder.

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Key Concept 3: The Experiment (The Toric Code)

To prove this, the authors used a famous model called the Toric Code.

• The Setup: Imagine a giant checkerboard where the rules are very specific.
• The Mess: They introduced "disorder" (randomness) into the rules. Sometimes the rules are slightly off, like a checkerboard where some squares are slightly warped.
• The Outcome: They found that depending on how much warping there was, the system fell into three different buckets:
  1. Perfect Order: The warping is tiny; the secret code is intact.
  2. Intrinsically Mixed (The Star of the Show): The warping is moderate. The strict rules break down into weak rules. The system enters the "SW-SSB" phase. It has a "classical memory" (you can store information, but only in a specific, robust way).
  3. Total Chaos: The warping is too high; the secret code is gone, and it's just random noise.

Why Does This Matter?

1. New Physics: It shows that "mixed states" (which describe real-world materials that are never perfectly isolated) have their own unique phases of matter that we didn't know about before.
2. Quantum Computing: In quantum computers, errors (noise) are a huge problem. This paper suggests that even with errors, there might be a "safe zone" (the SW-SSB phase) where information can still be stored and protected, even if the system isn't perfectly pure. It's like finding a way to keep a secret safe even if the safe door is slightly ajar.
3. Refining the Map: It redraws the map of the quantum world. Previously, we thought there were only "Perfect" and "Broken" states. Now we know there is a whole new continent of "Mixed" states in between.

Summary Analogy

Imagine a dance floor.

• Pure State: Everyone is dancing in a perfect, synchronized line.
• Trivial State: Everyone is dancing randomly, bumping into each other.
• Intrinsically Mixed (SW-SSB): The music is slightly distorted. The dancers can't keep the perfect line anymore (Strong symmetry breaks), but they don't just stop dancing. Instead, they form a new, looser pattern where they move in groups based on a general rhythm (Weak symmetry). This new group dance is stable and unique. You can't get it by just slowing down the perfect dance; you need the distortion to create it.

The paper is the guidebook that finally explains the rules of this new, looser dance.

Technical Summary

1. Problem Statement

The paper addresses the classification of mixed-state topological orders in 2+1 dimensions. While pure-state topological orders are well-understood as phases with spontaneously broken 1-form symmetries, the definition of phases for mixed states (described by density matrices $\rho$) remains subtle.

• The Limitation of Current Equivalence Relations: The standard equivalence relation for mixed states is two-way finite-depth channel connectivity. Two states are in the same phase if they can be connected by finite-depth local quantum channels in both directions. Under this definition, many "intrinsically mixed" topological orders (states that cannot be purified to a pure topological order) are deemed trivial because they are connected to the infinite-temperature (completely mixed) state.
• The Phenomenon: Recent work has identified Strong-to-Weak Spontaneous Symmetry Breaking (SW-SSB) in mixed states. In SW-SSB, a symmetry is "strong" (acting as $U\rho \propto \rho$) in the pure limit but becomes only "weak" (acting as $U\rho U^\dagger = \rho$) or trivial in the mixed state, or vice versa. Specifically, the authors focus on 1-form symmetries where the strong symmetry breaks down to a weak one, creating a distinct phase that standard equivalence relations fail to distinguish from trivial states.

2. Methodology

The authors propose a refined equivalence relation and apply it to disordered ensembles of topological systems.

A. Refined Equivalence Relation: Finite Rényi-2 Markov Length

Instead of relying solely on channel connectivity, the authors define two density matrices $\rho_A$ and $\rho_B$ to be in the same phase if and only if:

1. They are connected by a finite-time Lindbladian evolution (continuous time quantum channel).
2. Along this path, the Rényi-2 Markov length remains finite and analytically varying.

• Rényi-2 Markov Length ($\xi_2$): This is defined via the decay of the Rényi-2 Conditional Mutual Information (CMI), $I_2(A:C|B)$.
• Connection to Choi States: The authors demonstrate that the Rényi-2 CMI of a mixed state $\rho$ is directly related to the connected correlation functions of its Choi state $|\rho\rangle\rangle$ (a pure state in a doubled Hilbert space).
  • If the Choi state undergoes a phase transition (diverging correlation length), the Rényi-2 Markov length of $\rho$ diverges.
  • This allows the classification of mixed-state phases to map onto the classification of pure-state phases of their corresponding Choi states.
• Advantage: This relation is "finer" than two-way channel connectivity. It distinguishes states that are channel-connected but have different underlying topological structures in their Choi representations (e.g., distinguishing intrinsically mixed states from trivial states).

B. Strong vs. Weak 1-Form Symmetries

The paper formalizes symmetry breaking patterns for mixed states:

• Strong Symmetry: $U_g \rho = e^{i\theta} \rho$.
• Weak Symmetry: $U_g \rho U_g^\dagger = \rho$.
• SW-SSB: A phase where a strong symmetry in the pure limit becomes a weak symmetry (or breaks differently) in the mixed state, leading to "intrinsically mixed" topological orders (non-modular anyon theories).

C. Model Systems

The authors investigate two primary examples of disordered toric code ensembles:

1. Random Vertex Terms: A toric code Hamiltonian with random coefficients on the vertex terms ($A_v$).
2. Random Field: A toric code Hamiltonian with random magnetic fields ($h_e Z_e$) added to the links.

3. Key Contributions and Results

A. Identification of Distinct Phases

Using the Rényi-2 Markov length criterion, the authors identify three distinct phases in the disordered toric code ensembles:

1. ST-SSB (Strong-to-Trivial): Both strong and weak 1-form symmetries are spontaneously broken. The Choi state corresponds to two copies of the toric code (pure topological order).
2. SW-SSB (Strong-to-Weak): The strong Wilson symmetry breaks down to a weak symmetry, while the 't Hooft symmetry remains weak but broken. The Choi state corresponds to a single copy of the toric code. This is the "intrinsically mixed" phase.
3. WS (Weakly Symmetric): All symmetries are weak and unbroken. The Choi state is trivial (infinite temperature).

B. Analysis of Random Vertex Terms (Gaussian Disorder)

• For a toric code with random vertex coefficients drawn from a Gaussian distribution, the resulting density matrix $\rho(\Delta_A)$ is exactly solvable.
• Result: For any non-zero disorder strength ($\Delta_A > 0$), the system enters the SW-SSB phase.
• Physical Interpretation: This phase is equivalent to a toric code at finite temperature where the plaquette terms are infinitely strong. It possesses a topological conditional mutual information (LW CMI) of $\log 2$ (half that of the pure toric code), confirming it is an intrinsically mixed state.
• Robustness: Perturbative calculations show this phase exists over a finite parameter range, proving it is a stable phase under the new equivalence relation.

C. Analysis of Random Fields

• Bimodal Distribution: The system exhibits three distinct regimes depending on the disorder strength $h$ relative to the toric code couplings ($\lambda_A, \lambda_B$):
  • $h \ll \lambda$: ST-SSB (Toric code phase).
  • $\lambda_A \ll h \ll \lambda_B$: SW-SSB (Intrinsically mixed phase).
  • $h \gg \lambda$: WS (Trivial phase).
• Gaussian Distribution: The behavior differs qualitatively. There is no exact strong symmetry for any finite disorder, but an emergent strong symmetry exists perturbatively. The system still exhibits the SW-SSB phase, characterized by specific scaling of order parameters ($O_{1,z} \sim e^{-\mu r/\xi}$).
• Duality: The authors map the random field toric code to a 2D random-bond Ising model in a transverse field. The transition from the ST-SSB phase to the SW-SSB phase in the toric code is dual to the paramagnet-to-spin-glass transition in the Ising model.

D. Decohered Toric Code

The paper revisits the decohered toric code (incoherent noise).

• They show that the "classical memory" phase (where $p > p_c$) is an SW-SSB phase.
• Under the standard two-way channel connectivity, this phase is trivial (connected to infinite temperature).
• Under the Rényi-2 Markov length criterion, it is a distinct phase from the infinite temperature state, with a critical error rate $p_c^{(2)} \approx 0.178$ (distinct from the von Neumann threshold $p_c^{(1)} \approx 0.109$).

4. Significance

1. New Classification Framework: The paper establishes a rigorous, finer classification of mixed-state phases using Rényi-2 Markov length. This resolves the ambiguity of "intrinsically mixed" topological orders, which were previously considered trivial under standard definitions.
2. Intrinsically Mixed Topological Order: It provides a concrete physical realization of "intrinsically mixed" orders (non-modular anyon theories) arising naturally from disordered ensembles, rather than just being abstract constructs.
3. SW-SSB as a Universal Phenomenon: The work demonstrates that Strong-to-Weak symmetry breaking is a robust mechanism for generating mixed-state topological order, observable in both decoherence and disorder scenarios.
4. Connection to Spin Glasses: The duality between the toric code with random fields and the transverse-field Ising spin glass provides a powerful tool for understanding mixed-state transitions using established results from spin glass theory.
5. Experimental Relevance: The results suggest that mixed-state topological order is robust and can be engineered or observed in systems with disorder or measurement errors, provided the correct diagnostic (Rényi-2 CMI/Markov length) is used.

In summary, the paper redefines the landscape of mixed-state quantum phases by introducing a more sensitive equivalence relation, successfully identifying and characterizing stable "intrinsically mixed" phases driven by the strong-to-weak breaking of 1-form symmetries in disordered topological systems.