  A microscopic calculation of ground state entanglement for the XY and
Heisenberg models shows the emergence of universal scaling behavior at quantum
phase transitions. Entanglement is thus controlled by conformal symmetry. Away
from the critical point, entanglement gets saturated by a mass scale. Results
borrowed from conformal field theory imply irreversibility of entanglement loss
along renormalization group trajectories. Entanglement does not saturate in
higher dimensions which appears to limit the success of the density matrix
renormalization group technique. A possible connection between majorization and
renormalization group irreversibility emerges from our numerical analysis.
