  Since the work of Ryu and Takayanagi, deep connections between quantum
entanglement and spacetime geometry have been revealed. The negative
eigenvalues of the partial transpose of a bipartite density operator is a
useful diagnostic of entanglement. In this paper, we discuss the properties of
the associated entanglement negativity and its R\'enyi generalizations in
holographic duality. We first review the definition of the R\'enyi
negativities, which contain the familiar logarithmic negativity as a special
case. We then study these quantities in the random tensor network model and
rigorously derive their large bond dimension asymptotics. Finally, we study
entanglement negativity in holographic theories with a gravity dual, where we
find that R\'enyi negativities are often dominated by bulk solutions that break
the replica symmetry. From these replica symmetry breaking solutions, we derive
general expressions for R\'enyi negativities and their special limits including
the logarithmic negativity. In fixed-area states, these general expressions
simplify dramatically and agree precisely with our results in the random tensor
network model. This provides a concrete setting for further studying the
implications of replica symmetry breaking in holography.
