Symmetry-resolved genuine multi-entropy: Haar random and graph states

We study the symmetry-resolved genuine multi-entropy, a measure that captures genuine multi-partite entanglement, in Haar random states and random graph states in the presence of a conserved quantity. For Haar random states, we derive explicit formulae for the genuine multi-entropy under a global $U(1)$ symmetry in the thermodynamic limit, and find that its dependence on subsystem sizes closely resembles that of fully Haar random states without a conserved charge. We also perform numerical analyses, focusing on spin systems for both Haar random and graph states. For random graph states, our numerical analyses reveal distinctive features of their multi-partite entanglement structure and we contrast them with the Haar random case.