Defect Verlinde Formula for Edge Excitations in Topological Order

Area of Science:

• Condensed matter physics
• Topological quantum matter

Background:

• Topological orders in 2+1 dimensions exhibit exotic properties.
• Boundary excitations and defects are crucial for understanding topological phases.

Purpose of the Study:

• To derive a general formula for boundary excitations in nonchiral bosonic topological orders.
• To establish a connection between fusion rules and topological properties like half-linking numbers.
• To explore applications in quantum computing.

Main Methods:

• Physical considerations to derive a formula analogous to the Verlinde formula.
• Calculation of half-linking numbers for condensed anyons and boundary excitations.
• Explicit computation for Abelian and non-Abelian topological orders.

Main Results:

• A formula relating fusion rules of boundary excitations to the half-linking number is derived.
• The half-linking number quantifies the interaction between condensed anyons and confined boundary excitations.
• Demonstrated computation of these numbers in concrete examples.

Conclusions:

• The derived formula provides a fundamental insight into topological orders and their boundaries.
• The results have potential applications in designing and realizing quantum computing devices.
• Understanding boundary phenomena is key to harnessing topological properties for quantum information processing.