Symmetry-resolved modular correlation functions in free fermionic theories

TL;DR

This paper introduces a framework for analyzing symmetry-resolved modular correlation functions in theories with a U(1) symmetry, providing tools for free fermionic models and revealing charge equipartition in correlations.

Contribution

It defines symmetry-resolved modular flow and correlation functions, and applies these concepts to free fermionic theories, highlighting charge equipartition in the correlations.

Findings

Modular correlation functions are independent of charge sector at leading order.

The framework satisfies the KMS condition in each symmetry sector.

Provides a computational toolkit for free fermionic theories.

Abstract

As a new ingredient for analyzing the fine structure of entanglement, we study the symmetry resolution of the modular flow of $U(1)$-invariant operators in theories endowed with a global $U(1)$ symmetry. We provide a consistent definition of symmetry-resolved modular flow that is defined for a local algebra of operators associated to a sector with fixed charge. We also discuss the symmetry-resolved modular correlation functions and show that they satisfy the KMS condition in each symmetry sector. Our analysis relies on the factorization of the Hilbert space associated to spatial subsystems. We provide a toolkit for computing the symmetry-resolved modular correlation function of the charge density operator in free fermionic theories. As an application, we compute this correlation function for a $1+1$-dimensional free massless Dirac field theory and find that it is independent of the charge sector at leading order in the ultraviolet cutoff expansion. This feature can be regarded as a charge equipartition of the modular correlation function. Although obtained for free fermions, these results may be of potential interest for bulk reconstruction in AdS/CFT.