  In this paper we examine how the predictions of conformal invariance can be
widely exploited to overcome the difficulties of the density-matrix
renormalization group near quantum critical points. The main idea is to match
the set of low-lying energy levels of the lattice Hamiltonian, as a function of
the system's size, with the spectrum expected for a given conformal field
theory in two dimensions. As in previous studies this procedure requires an
accurate targeting of various excited states. Here we discuss how this can be
achieved within the DMRG algorithm by means of the recently proposed
Thick-restart Lanczos method. As a nontrivial benchmark we use an anisotropic
spin-1 Hamiltonian with special attention to the transitions from the Haldane
phase. Nonetheless, we think that this procedure could be generally valid in
the study of quantum critical phenomena.
