  Entanglement entropy and entanglement spectrum have been widely used to
characterize quantum entanglement in extended many-body systems. Given a pure
state of the system and a division into regions $A$ and $B$, they can be
obtained in terms of the $Schmidt~ values$, or eigenvalues $\lambda_{\alpha}$
of the reduced density matrix $\rho_A$ for region $A$. In this paper we draw
attention instead to the $Schmidt~ vectors$, or eigenvectors
$|v_{\alpha}\rangle$ of $\rho_A$. We consider the ground state of critical
quantum spin chains whose low energy/long distance physics is described by an
emergent conformal field theory (CFT). We show that the Schmidt vectors
$|v_{\alpha}\rangle$ display an emergent universal structure, corresponding to
a realization of the Virasoro algebra of a boundary CFT (a chiral version of
the original CFT). Indeed, we build weighted sums $H_n$ of the lattice
Hamiltonian density $h_{j,j+1}$ over region $A$ and show that the matrix
elements $\langle v_{\alpha}H_n |v_{\alpha'}\rangle$ are universal, up to
finite-size corrections. More concretely, these matrix elements are given by an
analogous expression for $H_n^{\tiny \text{CFT}} = \frac 1 2 (L_n + L_{-n})$ in
the boundary CFT, where $L_n$'s are (one copy of) the Virasoro generators. We
numerically confirm our results using the critical Ising quantum spin chain and
other (free-fermion equivalent) models.
