Defect Relative Entropy

TL;DR

This paper introduces the concept of defect relative entropy in conformal field theories, providing formulas and calculations for various models, revealing zero entropy between certain topological defects and defining the defect relative sector.

Contribution

It presents the first comprehensive formulation of defect relative entropy, including universal and model-specific expressions, and explores its implications in topological defect classification.

Findings

Derived a universal formula reducing to Kullback-Leibler divergence.

Calculated defect relative entropy for models like Ising and WZW.

Discovered zero defect relative entropy between certain topological defects.

Abstract

We introduce the concept of \textit{defect relative entropy} as a measure of distinguishability within the space of defects. We compute the defect relative entropy for conformal/topological defects, deriving a universal formula in conformal field theories (CFTs) on a circle. This formula reduces to the Kullback-Leibler divergence. Furthermore, we provide a detailed expression of the defect relative entropy for diagonal CFTs with specific topological defect choices, utilizing the theory's modular $\mathcal{S}$ matrix. We also present a general formula for the \textit{ defect sandwiched R\'enyi relative entropy} and the \textit{defect fidelity}. Through explicit calculations in specific models, including the Ising model, the tricritical Ising model, and the $\widehat{su}(2)_{k}$ WZW model, we have made an intriguing finding: zero defect relative entropy between reduced density matrices associated with certain topological defect. Notably, we introduce the concept of the \textit{defect relative sector}, representing the set of topological defects with zero defect relative entropy.