  We study the momentum space entanglement spectra of bosonic and fermionic
formulations of the spin-1/2 XXZ chain with analytical methods and exact
diagonalization. We investigate the behavior of the entanglement gaps, present
in both partitions, across quantum phase transitions in the XXZ chain. In both
cases, finite size scaling reveals that the entanglement gap closure does not
occur at the physical transition points. For bosons, we find that the
entanglement gap observed in [Thomale et al., Phys. Rev. Lett. 105, 116805
(2010)] depends on the scaling dimension of the conformal field theory as
varied by the XXZ anisotropy. For fermions, the infinite entanglement gap
present at the XX point persists well past the phase transition at the
Heisenberg point. We elaborate on how these shifted transition points in the
entanglement spectra may in fact support the numerical study of physical
transitions in the momentum space density matrix renormalization group.
