Quantum Phase Transitions in Anisotropic Spin-1 Chains: A Density-Matrix Renormalisation Group Approach

According to Haldane's conjecture, the energy gap in the spectrum of excitations of quantum spin chains depends on the actual value of the local magnetic moments: It vanishes for half-integer spins while it remains finite for integer values. This genuine quantum phenomenon extends to spin ladders where a central role is played also by the number of legs. Nowadays such features have received a number of experimental confirmations and one can even build quantitative models of these low-dimensional magnetic systems by adding different types of anisotropies to the basic Heisenberg Hamiltonian. The natural questions that arise are how robust the Haldane phase is and whether these anisotropic terms can induce a closure of the gap, that is, a quantum phase transition. \\The density-matrix renormalisation group (DMRG) method works remarkably well far from critical regions and provides an accurate numerical description of systems with hundreds of sites. One would like to exploit this accuracy on these sizes to approach quantum critical points, but there are some difficulties related to the appearance of a ``fake'' characteristic length due to the truncation error inherent to the DMRG. Here we discuss how to circumvent these problems by means of a suitable combination of the DMRG and finite-size scaling techniques. We drive special attention to nontrivial transitions involving a change in the hidden topological order typical of the Haldane phase. The analysis benefits from exact results of conformal field theories and eventually we are able to classify different types of transition by inspecting the structure of the spectrum of the low-lying excitations and by extracting the values of the critical indices from the various kinds of correlation functions that can be computed within the DMRG scheme.