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.

9 pages, 3 figures; v2: added explicit results for a generic ground state of the toric code, and extended comparison with previous literature; v3: adds numerical calculations for the entanglement asymmetry, and matches published text

References in corpus (37)

• Anyons in an exactly solved model and beyond
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Topological quantum memory
• Generalized Global Symmetries
• Coupling a QFT to a TQFT and Duality
• Theta, Time Reversal, and Temperature
• Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)
• Generalized Symmetries in Condensed Matter
• Quasi-particle Statistics and Braiding from Ground State Entanglement
• A symmetry principle for Topological Quantum Order
• Ground state entanglement and geometric entropy in the Kitaev's model
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids
• Symmetry and Topological Order
• Entanglement asymmetry as a probe of symmetry breaking
• Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems
• A quantum topological phase transition at the microscopic level
• Microscopic origin of the quantum Mpemba effect in integrable systems
• Symmetry restoration and quantum Mpemba effect in symmetric random circuits
• Entanglement asymmetry and quantum Mpemba effect in the XY spin chain
• Spurious topological entanglement entropy from subsystem symmetries
• Goldstone modes and photonization for higher form symmetries
• Entanglement asymmetry and quantum Mpemba effect in two-dimensional free-fermion systems
• Spurious Long-range Entanglement and Replica Correlation Length
• Lack of symmetry restoration after a quantum quench: an entanglement asymmetry study
• An Introduction to Higher-Form Symmetries
• Interactions along an Entanglement Cut in 2+1D Abelian Topological Phases
• Entangled multiplets, asymmetry, and quantum Mpemba effect in dissipative systems
• Entanglement Entropy of U(1) Quantum Spin Liquids
• A universal formula for the entanglement asymmetry of matrix product states
• Entanglement asymmetry in CFT and its relation to non-topological defects
• $\theta =\pi$ in $SU(N)/{\mathbb{Z}}_N$ gauge theories
• Universal lower bound on topological entanglement entropy
• Entanglement asymmetry in conformal field theory and holography
• Entanglement Properties of Gauge Theories from Higher-Form Symmetries
• Gaining insights on anyon condensation and 1-form symmetry breaking across a topological phase transition in a deformed toric code model