Non-coplanar Model States in Quantum Magnetism Applications of the High-Order Coupled Cluster Method

Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. Here we extend the CCM to non-coplanar / 3D model states and we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-$S$ triangular-lattice {\it XXZ} antiferromagnet in the presence of an external magnetic field, for the cases $\frac{1}{2} \leq S \leq5 $. For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required. Excellent correspondence is seen with the results of other methods, where they exist, for these systems. CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic {\it XXZ} model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the {\it XXZ} anisotropy parameter $\Delta$) between a 3D ground state and a coplanar ground state for the {\it XXZ} model for values for the external magnetic field near to saturation, for values of the spin quantum number $S \leq 5$.