We consider a quasi-one-dimensional system of spin-1 Heisenberg antiferromagnetic chains in two-dimensional and three-dimensional hypercubic lattices with interchain coupling J and uniaxial single-ion anisotropy D. Using large-scale numerical simulations, we map out the J-D phase diagram and investigate the low-lying excitations of the Haldane phase in the J≪1 limit. We also provide direct evidence that the Haldane phase remains a nontrivial symmetry-protected topological state for small but finite J.

We study the nature of melting of a two dimensional (2D) Lennard-Jones solid using large scale Monte Carlo simulation. We use systems of up to 102,400 particles to capture the decay of the correlation functions associated with translational order (TO) as well as the bond-orientational (BO) order. We study the role of dislocations and disclinations and their distribution functions. We computed the temperature dependence of the second moment of the TO order parameter (ΨG) as well as of the order parameter Ψ6 associated with BO order. Applying finite-size scaling of these second moments we determined the anomalous dimension critical exponents η(T ) and η6(T ) associated with power-law decay of the ΨG and Ψ6 correlation functions. We also computed the temperature dependent distribution of the order parameters ΨG and Ψ6 on the complex plane which support a two stage melting with a hexatic phase as an intermediate phase. From the correlation functions of ΨG and Ψ6 we extracted the corresponding temperature dependent correlation lengths ξ(T ) and ξ6(T ). The analysis of our results leads to a consistent picture strongly supporting a two stage melting scenario as predicted by the Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY) theory where melting occurs via two continuous phase transitions, first from solid to a hexatic fluid at temperature Tm, and then from the hexatic fluid to an isotropic fluid at a critical temperature Ti. We find that ξ(T ) and ξ6(T ) have a distinctly different temperature dependence each diverging at different temperature and that their finite size scaling properties are consistent with the KTHNY theory. We also used the temperature dependence of η and η6 and their theoretical bounds to provide estimates for the critical temperatures Tm and Ti, which can also be estimated using the Binder ratio. Our results are within error bars the same as those extracted from the divergence of the correlation lengths.

We use a generalized spin wave approach and large scale quantum Monte Carlo (QMC) simulations to study the quantum phase diagram and quasiparticle excitations of the S = 1 Heisenberg model with an easy-plane single-ion anisotropy in dimensions d = 2 and 3. We consider two alternative approximations for describing the quantum paramagnetic state: the standard Holstein-Primakoff approximation and a modified treatment in which the local constraint (finite dimension of the local Hilbert space) is enforced by introducing a Lagrange multiplier. While both approximations produce qualitatively similar results, the latter approach is the only one that is in good quantitative agreement with the quantum phase diagram and the quasiparticle dispersions obtained with QMC. This result is very important for low-temperature studies of quantum paramagnets in magnetic fields because it shows that a simple modification of the standard analytical approach should produce much better quantitative agreement between theory and experiment.

We review the basic properties of the Haldane phase in spin-1 Heisenberg antiferromagnetic chains, including its persistence in quasi-one-dimensional geometries. Using large-scale numerical simulations, we map out the phase diagram for a realistic model applicable to experimental Haldane compounds. We also investigate the effect of different chain coupling geometries and confirm a general mean field universality of the critical coupling times the coordination number of the lattice. Inspired by the recent development of characterization of symmetry protected topological states, of which the Haldane phase of spin-1 Heisenberg antiferromagnetic chain is a preeminent example, we provide direct evidence that the quasi-one-dimensional Haldane phase is indeed a non-trivial symmetry protected topological state.