More on symmetry resolved operator entanglement

The `operator entanglement' of a quantum operator O is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global U(1) conservation law, and on operators O with a well-defined U(1) charge, for which it is possible to resolve the operator entanglement of O according to the U(1) symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in [PRX Quantum 4, 010318 (2023)] and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix ρ_β= e^{- βH} and of charged local operators evolving in Heisenberg picture O = e^{i t H} O e^{-i t H}. Our main results are: i) the SROE of ρ_β obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.