Operator Content of Critical Anisotropic Spin-1 Chains: Combined Results from DMRG and Planar Nonlinear Sigma Model

We study the critical regions in the phase diagram of spin-1 chains with Ising-like and single-site anisotropies, with special emphasis on the transitions from the Haldane phase. On the numerical side, we use a DMRG scheme suitably extended to handle multiple excited states in different spin sectors. At the same time, we can compute two-points and string-order correlation functions on the ground state and derive the associated decay exponents by means of finite-size scaling arguments. As a result, we reconstruct the whole spectrum of the energy levels associated with relevant operators at the critical lines establishing a precise connection with the predictions of conformal field theories. This suggets a revisitation of the structure and the origin of the so-called ``critical-fan'' in connection with the microscopic spin Hamiltonian. In particular, as far as the Haldane-Large $D$ transition is concerned, we discuss the mapping onto an O(2) nonlinear $\sigma$ model that correctly predicts the pure Gaussian operator content observed along the line and also provides quite good estimates of the compactification radius as a function of the anisotropy parameters.