Braiding statistics of loop excitations in three dimensions.

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory
• Topological Phases of Matter

Background:

• Three-dimensional quantum many-body systems exhibit complex excitation braiding statistics.
• Existing studies focus on particle-particle, particle-loop, or loop-loop braiding.
• The role of linked loops in braiding statistics remains less explored.

Purpose of the Study:

• To investigate the fundamental nature of three-loop braiding statistics.
• To determine if three-loop braiding can differentiate between various symmetry-protected topological phases.
• To analyze three-loop braiding within (Z(N))(K) gauge theories derived from lattice boson models.

Main Methods:

• Formulation of (Z(N))(K) gauge theories by gauging gapped, short-range entangled lattice boson models.
• Analysis of the statistical phase generated by braiding one loop around another, linked by a third loop.
• Comparison of braiding statistics across different symmetry-protected topological phases with identical (Z(N))(K) symmetry.

Main Results:

• Identified a fundamental quantity: the statistical phase from braiding one loop around another, linked by a third.
• Demonstrated that three-loop braiding statistics are sensitive to the underlying topological order.
• Showcased the ability of three-loop braiding to distinguish between distinct short-range entangled bosonic states sharing the same (Z(N))(K) symmetry.

Conclusions:

• Three-loop braiding statistics offer a more fundamental characterization of topological phases than pairwise braiding.
• This braiding measure provides a powerful tool for classifying and distinguishing symmetry-protected topological phases.
• The findings advance the understanding of topological order and quantum entanglement in many-body systems.