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

Emergent universality in critical quantum spin chains: entanglement Virasoro algebra