Finite-Size Analysis of DMRG Data to Approach Quantum Critical Points: The Example of Anisotropic Spin-1 Chains

Despite its name, the density-matrix renormalisation group (DMRG) is a numerical technique born to study, with very high accuracy, quantum systems far from criticality where the excitation spectrum has a well-defined energy gap. On the one hand, with current machines one can deal with spin or electronic systems having hundreds of sites. On the other hand, it is known that there are still some difficulties in exploiting these performances near critical points. Essentially these can be traced back to the appearance of a fake characteristic length as a consequence of the main (and maybe unique) source of approximation in the DMRG, namely the truncation of the Hilbert space to the $m$ states with largest eigenvalues of the density matrix. Here we approach this problem with a combined analysis of the DMRG data both with elements of finite-size scaling theory and predictions from conformal field theories. In particular, the latter allow for a characterisation of the full energy spectrum, once that a suitable number of levels is computed. In this sense, we present the results of a DMRG code whose Lanczos routine can target various excited states with high precision by means of the so-called thick-restart algorithm. An accurate multi-target scheme like this may be useful also for the calculation of thermal and dynamical properties of the system. These points are illustrated taking as a reference model the Hamiltonian of spin-1 chains with Ising-like and single-site anisotropies. Special emphasis is put on the transitions from the Haldane phase, involving a change in the hidden topological order (Z$_2 \times$Z$-2$ symmetry breaking). Nonetheless, the methods presented here may prove to be useful also for the analysis of spatially-finite low-dimensional quantum systems with nontrivial boundary conditions.