Higher-form entanglement asymmetry and topological order

1. Problem Statement

The paper addresses the challenge of detecting topological order in quantum many-body systems, specifically focusing on Abelian topological phases in two dimensions.

• Context: Traditional Landau symmetry-breaking theory classifies phases via local order parameters. However, topological order lies outside this paradigm. Recent developments have reinterpreted Abelian topological order as the spontaneous symmetry breaking (SSB) of higher-form symmetries (specifically 1-form symmetries in 2D).
• The Gap: While 1-form SSB explains the ground state degeneracy of topological phases, it is unclear if this breaking can be detected locally within a subregion. Standard order parameters are local, but 1-form symmetries act on extended objects (loops/strings).
• Specific Question: Can a measure of symmetry breaking, specifically entanglement asymmetry, be generalized to higher-form symmetries to serve as a local probe for topological order? Furthermore, how does this relate to the Topological Entanglement Entropy (TEE), the standard metric for topological order?

2. Methodology

The authors generalize the concept of entanglement asymmetry (originally defined for 0-form symmetries) to 1-form symmetries.

• Definition of Higher-Form Entanglement Asymmetry ($\Delta S_A$):

  • For a subregion $A$ and a symmetry operator $U_\gamma$ supported on a loop $\gamma$, the reduced density matrix is $\rho_A = \text{Tr}_B(|\psi\rangle\langle\psi|)$.
  • The symmetry operator factorizes as $U_\gamma = U_{\gamma, A} \otimes U_{\gamma, B}$.
  • The symmetrized density matrix is constructed by averaging over the symmetry group $G$:
    $$\rho_A^{\text{sym}} = \frac{1}{|G|} \sum_{g \in G} U_{\gamma, A}(g) \rho_A U_{\gamma, A}^\dagger(g)$$
  • The asymmetry is the relative entropy (difference in von Neumann entropy):
    $$\Delta S_A = S[\rho_A^{\text{sym}}] - S[\rho_A]$$
  • Crucial Constraint: Because 1-form operators are topological, $\Delta S_A$ is non-zero only if both the subregion $A$ and the symmetry support $\gamma$ are non-contractible (e.g., wrapping around a torus). Contractible regions yield $\Delta S_A = 0$.

• Model Systems:

  1. Toric Code: Used as the primary example of Abelian topological order with a $\mathbb{Z}_2 \times \mathbb{Z}_2$ 1-form symmetry (electric and magnetic flux conservation).
  2. Deformed Toric Code: A model where a deformation term breaks the magnetic 1-form symmetry (specifically the 't Hooft loop symmetry) while preserving the electric one, creating a phase transition between a topological phase and a non-topological phase.

• Calculation Strategy:

  • The authors compute $\Delta S_A$ for ground states of the toric code, specifically analyzing Minimum Entropy States (MES) and general superpositions.
  • They compare the scaling of $\Delta S_A$ against the TEE ($\gamma$) in the thermodynamic limit ($L \to \infty$).

3. Key Contributions

A. Generalization of Entanglement Asymmetry

The paper successfully extends entanglement asymmetry to higher-form symmetries. It establishes that for 1-form symmetries, the asymmetry is a topological quantity dependent on the non-contractible nature of the subregion and the symmetry loop. It quantifies the "information" about global symmetry breaking accessible within a local subregion.

B. Relationship between Asymmetry and TEE

The authors clarify the relationship between the 1-form entanglement asymmetry and the Topological Entanglement Entropy (TEE):

• Connection: For the Toric Code on a non-contractible cylinder, the asymmetry $\Delta S_A$ extracts the sub-leading correction to the area law, which is exactly the TEE.
• Distinction: They demonstrate that while numerically equal in specific cases (MES), the quantities are conceptually distinct.
  • TEE arises from the constraint that flux strings must form closed loops (reducing entropy).
  • Asymmetry arises from the presence of different 1-form charge sectors (non-contractible strings) within the subregion.
• Uncertainty Principle: They derive a bound on the sum of asymmetries in orthogonal directions: $\Delta S_{A_x} + \Delta S_{A_y} \leq \log |G|$. This mirrors an uncertainty principle where maximizing symmetry breaking in one direction (via MES) forces symmetry preservation in the orthogonal direction due to the non-commutativity of Wilson and 't Hooft loops.

C. Probe for Topological Order in Deformed Models

The most significant result concerns the Deformed Toric Code:

• The Paradox: In the non-topological phase ($\beta > \beta_c$), the system still exhibits 1-form SSB (ground state degeneracy remains 4), yet it lacks topological order (TEE = 0).
• The Resolution: The authors show that while $\Delta S_A > 0$ at finite size in the non-topological phase, it vanishes in the thermodynamic limit ($L \to \infty$).
  • In the non-topological phase, the weight of states containing non-contractible loops inside the subregion $A$ becomes vanishingly small as $L$ increases. The reduced density matrix becomes effectively symmetric ($\rho_A \approx \rho_A^{\text{sym}}$).
  • In the topological phase, $\Delta S_A$ remains finite ($\approx \log 2$) as $L \to \infty$.
• Conclusion: The scaling behavior of entanglement asymmetry distinguishes true topological order from mere 1-form SSB.

4. Key Results

1. Toric Code Calculation: For a ground state like $|\Psi^Z_{00}\rangle$ on a torus with a cylindrical subregion $A$, the entanglement asymmetry is:
   $$\Delta S_A = \log 2$$
   This corresponds to one bit of information encoded in the symmetry breaking. For Minimum Entropy States (MES) that break both electric and magnetic symmetries, $\Delta S_A = 2 \log 2$.

2. Generalization to Abelian Order: For a general Abelian topological order with anyon species count $D$ (where the symmetry group size is $|G| = D$), the maximal entanglement asymmetry is:
   $$\Delta S_A^{\text{max}} = \log |G| = \log D$$
   This provides a direct link between the number of anyon species and the measurable asymmetry.

3. Deformed Toric Code Scaling:

   • Topological Phase ($\beta < \beta_c$): $\lim_{L \to \infty} \Delta S_A = \log 2$ (Non-zero).
   • Non-Topological Phase ($\beta > \beta_c$): $\lim_{L \to \infty} \Delta S_A = 0$.
   • This confirms that $\Delta S_A$ is a faithful probe of topological order, whereas simple ground state degeneracy is not.

5. Significance

• New Order Parameter: The paper proposes entanglement asymmetry as a robust, local order parameter for topological phases, capable of distinguishing them from trivial phases that share the same ground state degeneracy and 1-form SSB.
• Conceptual Clarity: It disentangles the concepts of "symmetry breaking" and "topological order," showing that while topological order can be viewed as 1-form SSB, the detection of this order requires specific scaling properties of the asymmetry.
• Anomaly Connection: The results suggest a deep link between the 1-form anomaly (the non-commutativity of Wilson and 't Hooft loops) and the persistence of entanglement asymmetry in the thermodynamic limit. The loss of the 't Hooft symmetry in the non-topological phase correlates with the vanishing of the asymmetry.
• Future Directions: The work opens avenues for studying non-invertible symmetries, continuous 1-form symmetries (e.g., U(1) gauge theories), and the dynamics of higher-form symmetry restoration (e.g., quantum Mpemba effects) in topological quantum computing contexts.

In summary, the paper provides a rigorous framework for using entanglement asymmetry to detect topological order, resolving ambiguities in systems where standard symmetry-breaking signatures persist even in the absence of topological order.