Abstract
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 λ_α of the reduced density matrix ρ_A for region A. In this paper we draw attention instead to the Schmidt vectors, or eigenvectors |v_α of ρ_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_α 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 v_αH_n |v_α' are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for H_n^ CFT = 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.