Fusion of Integrable Defects and the Defect $g$-Function

TL;DR

This paper investigates exact defect g-functions in 2D integrable quantum field theories to understand defect fusion, revealing additive properties for topological defects and oscillatory effects for non-topological ones.

Contribution

It provides a detailed analysis of defect fusion using g-functions, highlighting differences between topological and non-topological defects in integrable models.

Findings

Topological defect g-functions are additive and fusion is multiplicative.

Non-topological defects exhibit oscillatory finite-size effects due to phase factors.

Fusion with non-topological defects reduces the localized entropy contribution.

Abstract

We study exact defect $g$-functions for integrable line defects in two-dimensional integrable quantum field theory and use them to probe defect fusion. We consider three settings: fusion of purely transmitting topological defects, fusion of non-topological defects with reflection and transmission, and fusion of a defect with an integrable boundary. For topological defects, the separated logarithmic $g$-function is additive, and the fusion limit is controlled by the multiplicative composition of transmission factors. For non-topological defects, separation-dependent phases in the Bethe-Yang equations produce oscillatory finite-size effects, while the fused defect is described by effective reflection and transmission amplitudes. In the Ising examples studied here, fusion involving non-topological defects lowers the finite localized contribution to the entropy, whereas topological defect-boundary fusion leaves it unchanged.