Symmetry-resolved modular correlation functions in free fermionic theories

TL;DR

This work develops a symmetry-resolved framework for modular flow in theories with a global $U(1)$ symmetry by decomposing the local operator algebra into fixed-charge sectors and defining sector-specific modular operators and flows. The authors show that symmetry-resolved modular correlation functions satisfy the KMS condition in each sector and that the full, unresolved modular objects are recovered by summing over sectors with weights $p_V(q)$. They provide a practical toolkit for free-fermionic Gaussian states and demonstrate it in detail for a $1+1$-dimensional massless Dirac theory, where the leading UV behavior of the symmetry-resolved charge-density modular correlator is independent of the charge sector (equipartition), with sector dependence appearing only in subleading terms. The results have potential implications for AdS/CFT bulk reconstruction and stimulate further exploration of symmetry-resolved modular structures, including Connes-Radon-Nikodym flows and higher-spin currents in holographic contexts.

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.