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

Topological orders in 2+1d are spontaneous symmetry-breaking (SSB) phases of 1-form symmetries in pure states. The notion of symmetry is further enriched in the context of mixed states, where a symmetry can be either strong" or weak". In this work, we apply a Rényi-2 version of the proposed equivalence relation in [Sang, Lessa, Mong, Grover, Wang, & Hsieh, to appear] on density matrices that is slightly finer than two-way channel connectivity. This equivalence relation distinguishes general 1-form strong-to-weak SSB (SW-SSB) states from phases containing pure states, and therefore labels SW-SSB states as intrinsically mixed". According to our equivalence relation, two states are equivalent if and only if they are connected to each other by finite Lindbladian evolution that maintains continuously varying, finite Rényi-2 Markov length. We then examine a natural setting for finding such density matrices: disordered ensembles. Specifically, we study the toric code with various types of disorders and show that in each case, the ensemble of ground states corresponding to different disorder realizations form a density matrix with different strong and weak SSB patterns of 1-form symmetries, including SW-SSB. Furthermore we show by perturbative calculations that these disordered ensembles form stable phases" in the sense that they exist over a finite parameter range, according to our equivalence relation.

💡 Research Summary

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The paper introduces a refined equivalence relation for mixed‑state phases that goes beyond the standard two‑way finite‑depth channel connectivity. By employing the Rényi‑2 Markov length, the authors define two density matrices ρ_A and ρ_B to be equivalent only if there exists a finite‑time Lindbladian evolution connecting them while the Rényi‑2 Markov length remains finite and varies continuously throughout the process. This criterion is stricter than the von Neumann Markov length used in earlier works and allows the detection of phase boundaries that were previously invisible, in particular those associated with strong‑to‑weak spontaneous symmetry breaking (SW‑SSB) of 1‑form symmetries.

The authors first review the notion of strong and weak symmetries for higher‑form symmetries: a unitary U is a strong symmetry of a density matrix ρ if U ρ ∝ ρ, while it is a weak symmetry if U ρ U† = ρ. In pure‑state topological orders, 1‑form symmetries are always broken strongly and form a modular tensor category. In mixed states, however, the broken strong symmetries need only form a pre‑modular category, allowing degenerate S‑matrices. When a mixed state exhibits both strong and weak breaking of the same 1‑form symmetry, the authors label it “intrinsically mixed” and identify it with SW‑SSB.

A key technical tool is the Choi isomorphism: the Choi state |ρ⟩⟩ of a density matrix lives in a doubled Hilbert space and is pure. The Rényi‑2 Markov length of ρ coincides with the correlation length of its Choi state, so classifying mixed‑state phases under the new equivalence relation reduces to classifying the corresponding pure Choi states. The authors show that the Rényi‑2 Markov length diverges precisely when the Choi state develops long‑range correlations, which signals a phase transition.

To illustrate the framework, the paper studies the toric code (a Z₂ × Z₂ 1‑form symmetric model) under three types of disorder:

1. Random vertex terms – each vertex receives a random Z‑type field. Averaging over disorder realizations yields a density matrix ρ_V that simultaneously breaks the electric loop symmetry strongly and the magnetic string symmetry weakly. Perturbative calculations demonstrate that ρ_V remains in a stable SW‑SSB phase over a finite disorder strength range. The Rényi‑2 Markov length stays finite, and the Choi state exhibits non‑zero topological conditional mutual information (TCMI), confirming its intrinsically mixed nature.

2. Random plaquette terms – random magnetic fields on plaquettes preserve the electric loop symmetry strongly while weakly breaking the magnetic symmetry, realizing a strong‑to‑strong (ST‑SSB) phase. This phase is also stable under small disorder.

3. Random transverse field – both electric and magnetic random fields are applied. By tuning the disorder amplitude γ, three distinct phases appear:

   • ST‑SSB for γ < γ_{c1},
   • SW‑SSB for γ_{c1} < γ < γ_{c2},
   • Weakly symmetric (WS) for γ > γ_{c2}, where only weak symmetries survive. Perturbative RG analysis and an exact mapping to a decohered toric code at low disorder support the stability of each region. The SW‑SSB region is characterized by a finite Rényi‑2 Markov length, a non‑zero TCMI, and a Choi state whose topological entanglement entropy matches that of a pre‑modular category.

The authors also discuss why the SW‑SSB phase is “trivial” under the older two‑way channel connectivity: an explicit two‑way finite‑depth channel can connect the disordered toric‑code density matrix to the infinite‑temperature (completely mixed) state if one allows explicit breaking of the 1‑form symmetry. However, such a channel necessarily drives the Rényi‑2 Markov length to infinity, violating the refined equivalence condition. Hence the SW‑SSB phase is distinct in the new classification.

Additional technical results appear in the appendices: explicit construction of the two‑way channel linking the SW‑SSB state to the infinite‑temperature state, calculation of the Choi state’s topological mutual information, proof that Rényi‑2 locally indistinguishable sectors correspond to SW‑SSB, and a sequential circuit that prepares the purified state |ρ(β)⟩⟩ in logarithmic time.

In summary, the paper provides a concrete, computable criterion (finite Rényi‑2 Markov length under Lindbladian evolution) to distinguish intrinsically mixed topological orders from conventional pure‑state topological phases. By applying this framework to disordered toric‑code ensembles, the authors demonstrate that disorder can robustly generate strong‑to‑weak symmetry‑breaking phases, enriching the landscape of quantum memory and error‑correction models and opening new avenues for exploring higher‑form symmetries in open quantum systems.