Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\ensuremath{\alpha}$-R\'enyi entropy of a single interval for several critical quantum chains. We considered models with
U(1) symmetry such as the spin-1/2
XXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the models.
Based on the relationship between reduced and thermal density matrices in conformal field theory (CFT), we show that the entanglement spectrum of a conformal critical chain with exponentially decaying terms consists of conformal towers of the associated chiral CFT, with only weak finite-size effects. Through free-fermion and interacting examples, we show that these entanglement spectra present a reliable method to extract detailed CFT spectra from single wave functions without access to the parent Hamiltonian. We complement our method with a Wilsonian numerical renormalisation group algorithm for solving interacting, exponentially decaying chain Hamiltonians.