  Entanglement is a central feature of many-body quantum systems and plays a
unique role in quantum phase transitions.
  In many cases, the entanglement spectrum, which represents the spectrum of
the density matrix of a bipartite system, contains valuable information beyond
the sole entanglement entropy.
  Here we investigate the entanglement spectrum of the long-range XXZ model. We
show that within the critical phase it exhibits a remarkable self-similarity.
  The breakdown of self-similarity and the transition away from a Luttinger
liquid is consistent with renormalization group theory.
  Combining the two, we are able to determine the quantum phase diagram of the
model and locate the corresponding phase transitions. Our results are confirmed
by numerically-exact calculations using tensor-network techniques.
  Moreover, we show that the self-similar rescaling extends to the geometrical
entanglement as well as the Luttinger parameter in the critical phase.
  Our results pave the way to further studies of entanglement properties in
long-range quantum models.
