Higher-form entanglement asymmetry and topological order

Abstract

We extend a recently defined measure of symmetry breaking, the entanglement asymmetry, to higher-form symmetries. In particular, we focus on Abelian topological order in two dimensions, which spontaneously breaks a 1-form symmetry. Using the toric code as a primary example, we compute the entanglement asymmetry and compare it to the topological entanglement entropy. We find that while the two quantities are not strictly equivalent, both are sub-leading corrections to the area law and can serve as order parameters for the topological phase. We generalize our results to non-chiral Abelian topological order and express the maximal entanglement asymmetry in terms of the quantum dimension. Finally, we discuss how the scaling of entanglement asymmetry correctly detects topological order in the deformed toric code, where 1-form symmetry breaking persists even in a trivial phase.

Figures (3)

• Figure 1: A 1-form symmetry operator (red) and its string charges (blue) on a torus. Inside a cylindrical subregion $A$ (green), the total charge measured by the symmetry operator can take different values. The dark (light) red part of the symmetry operator represents its factorization into region A(B).
• Figure 2: A 1-form symmetry on a torus with trivial asymmetry, $\Delta S_A = 0$. (a) Contractible symmetry operator $U_\gamma$, (b) Contractible subregion $A$, (c) Symmetry-breaking in the horizontal direction (superposition of different numbers of blue charges) cannot be detected by a symmetry operator in the same direction, such as $U_\gamma$.
• Figure 3: In a toric code ground state, $\rho_A$ splits into sectors labeled by the pattern of string charge (blue curves) intersections at the boundary of $A$. Each sector splits into diagonal blocks of fixed charge in $A$, and off-diagonal blocks that mix charges in $A$. The structure of $\rho_A$ is insensitive to deformations of the symmetry operator (red).