Numerical and Analytical Investigation of the Operator Content of Anisotropic Spin-1 Chains Near Transition Lines

Although there are still some difficulties in exploiting the performances of the DMRG near critical points, there is general consensus on the fact that the possible universality classes of critical 1D quantum systems are well described by conformal field theories (CFT's). Once that the essential features of the appropriate CFT (central charge, degeneracies, ...) have been identified one can fruitfully employ the finite-size scaling predictions of the theory to extract the scaling dimensions and the critical exponents with high precision. However, from a numerical point of view, the identification of the correct CFT requires a careful analysis of various levels of the low-lying part of the spectrum. Taking as a nontrivial reference a two-parameter class of spin-1 Hamiltonians with anisotropic interactions along the z-axis, we discuss first the critical spectra obtained with the help of a multi-target DMRG code whose Lanczos routine can target various excited states at once by means of the so-called Thick-restart algorithm. We then focus on the off-critical continuum theories that one finds near the borders of the Haldane phase of the model, where a change of topological (hidden) order takes place. In particular, near the Gaussian (c=1) line we expect the appropriate field theory to be a massive sine-Gordon model and the numerical estimates of the exponents associated with the string order parameter (SOP) along z suggest that the corresponding correlator is related to the so-called twist operator. At the Ising-like (c=1/2) transition, instead, the longitudinal SOP remains finite while the transverse one vanishes, revealing some analogies with what happens in an isotropic spin-1 Heisenberg chain in a staggered magnetic field. Interestingly enough, spin systems like these, being characterised by a string-ordered ground state of valence-bond type, have recently been proposed as valid quantum channels for the high level of entanglement they can carry.