Strong-to-weak spontaneous symmetry breaking of higher-form non-invertible symmetries in Kitaev's quantum double model

1. Problem Statement

Topological orders (TO) are traditionally understood as phases of matter protected by long-range entanglement and anyonic excitations. Recent theoretical developments have reframed TOs as the result of Spontaneous Symmetry Breaking (SSB) of higher-form symmetries. However, most existing frameworks focus on pure states and Abelian symmetries.

In realistic physical systems, quantum states are inevitably coupled to environments, leading to decoherence and resulting in mixed states. The paper addresses two critical gaps:

1. How do non-invertible higher-form symmetries (specifically in non-Abelian systems) behave under decoherence?
2. What is the nature of the Strong-to-Weak Spontaneous Symmetry Breaking (SWSSB) mechanism in non-Abelian Kitaev quantum double models $D(G)$?

The authors aim to characterize the mixed-state phases of non-Abelian topological orders and determine how quantum information encoded in the ground state subspace degrades into classical information under local noise.

2. Methodology

The authors employ a combination of lattice gauge theory, quantum information theory, and category theory:

• Model System: They utilize the Kitaev Quantum Double Model $D(G)$ for a general finite non-Abelian group $G$ defined on a square lattice. The Hamiltonian is a sum of commuting vertex ($A_v$) and plaquette ($B_p$) projectors.
• Decoherence Channel: They introduce a specific local decoherence channel (a CPTP map) representing Z-type errors. This channel acts on each edge by incoherently proliferating electric charges corresponding to the irreducible representations (irreps) of the group $G$. The Kraus operators are constructed from generalized Pauli $Z$ operators ($Z_\Gamma$) associated with the representation category $\text{Rep}(G)$.
• Symmetry Analysis:
  • They define 1-form non-invertible symmetries using ribbon operators.
  • Strong Symmetries: Operators that commute exactly with the decohered density matrix $\rho_{cl}$.
  • Weak Symmetries: Operators that commute with the density matrix only on average (or satisfy specific commutation relations with the channel).
• Topological Invariants: They analyze the Information Convex Set and Locally Indistinguishable Sets to characterize the mixed state. They utilize the Conditional Mutual Information (CMI) to define the Markov length and prove local indistinguishability.
• Anomaly Detection: They investigate the 't Hooft anomaly between strong and weak ribbon symmetries to identify the symmetry breaking patterns.

3. Key Contributions and Results

A. Strong-to-Weak Spontaneous Symmetry Breaking (SWSSB)

The paper demonstrates that under Z-type decoherence, the non-Abelian quantum double model exhibits a distinct symmetry breaking pattern:

• Strong 1-form Symmetry Breaking: The strong 1-form non-invertible symmetries (generated by closed ribbon operators $F^\Gamma_\xi$) undergo Strong-to-Weak SSB (SWSSB). The symmetry is preserved in the mixed state but becomes "weak" in the sense that the order parameter (the ribbon operator) no longer commutes strictly with the state in the same way as in the pure state, or rather, the symmetry is broken in a way that distinguishes the mixed state from a trivial product state.
• Weak-to-Trivial SSB (WTSSB): Conversely, the weak 1-form symmetries (associated with magnetic fluxes/conjugacy classes) undergo Weak-to-Trivial SSB, meaning they are effectively broken down to trivial symmetries.
• Anomaly Structure: The authors show an 't Hooft anomaly between the strong and weak ribbon symmetries. When a closed ribbon is truncated to an open ribbon, it creates magnetic anyons at the endpoints. The strong symmetry detects these excitations, leading to the specific breaking pattern.
• Error Type Dependence: Crucially, they show that Z-type errors preserve the strong symmetry structure (associated with electric charges that braid trivially), whereas generic X-type errors would destroy all strong symmetries in non-Abelian models (except those in the group center).

B. Locally Indistinguishable Set and Information Convexity

The authors prove that the set of decohered ground states, denoted as $\mathcal{Q}\{\rho_{cl}\}$, forms a Locally Indistinguishable Set:

• Local Indistinguishability: For any simply connected subregion $A$, the reduced density matrices of all states in $\mathcal{Q}\{\rho_{cl}\}$ are identical ($\rho_A = \text{tr}_{\bar{A}}(\rho_{cl})$).
• Finite Markov Length: The conditional mutual information (CMI) decays exponentially with the width of the buffer region, indicating a finite Markov length.
• Information Convex Set: The set $\mathcal{Q}\{\rho_{cl}\}$ is a convex set. The authors prove that any convex combination of states in this set remains in the set.

C. Degradation of Quantum to Classical Information

A central result is the mapping of quantum information to classical information:

• Dimensionality Match: The number of extremal points in the information convex set $\mathcal{Q}\{\rho_{cl}\}$ is exactly equal to the ground-state degeneracy (GSD) of the original pure-state quantum double model.
• Isomorphism: The set of extremal points is isomorphic to the set of $G$-orbits of homomorphisms from the fundamental group of the manifold to the group $G$ under conjugation:
  $$\text{Ext}(\mathcal{Q}\{\rho_{cl}\}) \cong \text{Hom}(\pi_1(\Sigma), G) / \text{Ad}_G$$
• Physical Interpretation: The quantum superposition of different topological sectors (ground states) is destroyed by decoherence. The resulting mixed state is a classical statistical mixture of these sectors. The "quantum memory" is degraded into a "classical memory" stored in the convex set of density matrices.

D. Specific Examples

The authors provide explicit calculations for:

• $S_3$ Quantum Double: Demonstrating the 8-fold ground state degeneracy on a torus and how the fusion category $\text{Rep}(S_3)$ dictates the surviving strong symmetries.
• $D_4$ Quantum Double: Showing the 22-fold degeneracy and the specific structure of the decoherence channel preserving the $\text{Rep}(D_4)$ subcategory.

4. Significance and Outlook

• Generalization of Topological Order: This work extends the paradigm of topological order from pure states to mixed states, specifically addressing non-Abelian systems which were previously less understood in this context.
• New Phase Classification: It introduces a rigorous framework for classifying mixed-state topological phases based on SWSSB and Information Convex Sets, moving beyond traditional symmetry-protected topological (SPT) classifications.
• Quantum Error Correction Implications: The results suggest a fundamental limit on quantum memory in open systems. While the topological order (in the form of a locally indistinguishable set) persists, the quantum coherence is lost, leaving only classical information. This defines a "decodability transition" where the system shifts from storing quantum information to storing classical information.
• Future Directions: The authors propose that the error threshold for this transition coincides with the mixed-state phase transition between strong-to-trivial and strong-to-weak SSB. They also suggest extending this framework to String-net models and investigating the role of different error channels (e.g., X-type vs. Z-type) in non-Abelian systems.

In summary, the paper establishes that non-Abelian topological order under decoherence does not simply vanish but transforms into a classical topological phase characterized by a convex set of locally indistinguishable density matrices, governed by the strong-to-weak breaking of non-invertible higher-form symmetries.