Symmetry Protected Entanglement

Symmetry is an important property of quantum mechanical systems which may dramatically influence their behavior in and out of equilibrium. In this talk, we study the effect of symmetry on tripartite entanglement properties of typical states in  symmetric sectors of Hilbert space. In particular, we consider Abelian symmetries and derive explicit expression for the logarithmic entanglement negativity of systems with $\mathbb{Z}_N$ and $U(1)$ symmetry groups. To this end, we develop a diagrammatic method to incorporate partial transpose within random matrix theory of symmetric states and formulate a perturbation theory in the inverse of the Hilbert space dimension. We further present entanglement phase diagrams as the subsystem sizes are varied and show that there are qualitative differences between systems with and without symmetries.