  The study of entanglement spectra is a powerful tool to detect or elucidate
universal behaviour in quantum many-body systems. We investigate the scaling of
the entanglement (or Schmidt) gap $\delta\xi$, i.e., the lowest laying gap of
the entanglement spectrum, at a two-dimensional quantum critical point. We
focus on the paradigmatic quantum spherical model, which exhibits a
second-order transition, and is mappable to free bosons with an additional
external constraint. We analytically show that the Schmidt gap vanishes at the
critical point, although only logarithmically. For a system on a torus and the
half-system bipartition, the entanglement gap vanishes as $\pi^2/\ln(L)$, with
$L$ the linear system size. The entanglement gap is nonzero in the paramagnetic
phase and exhibits a faster decay in the ordered phase. The rescaled gap
$\delta\xi\ln(L)$ exhibits a crossing for different system sizes at the
transition, although logarithmic corrections prevent a precise verification of
the finite-size scaling. Interestingly, the change of the entanglement gap
across the phase diagram is reflected in the zero-mode eigenvector of the
spin-spin correlator. At the transition quantum fluctuations give rise to a
non-trivial structure of the eigenvector, whereas in the ordered phase it is
flat. We also show that the vanishing of the entanglement gap at criticality
can be qualitatively but not quantitatively captured by neglecting the
structure of the zero-mode eigenvector.
