Magnetic order in a spin-half interpolating square-triangle Heisenberg antiferromagnet

Using the coupled cluster method we study the zero-temperature phase diagram of a spin-half Heisenberg antiferromagnet (HAF), the so-called $J_{1}$--$J_{2}'$ model, defined on an anisotropic 2D lattice. With respect to an underlying square-lattice geometry the model contains antiferromagnetic ($J_{1} > 0$) bonds between nearest neighbors and competing ($J_{2}'>0$) bonds between next-nearest neighbors across only one of the diagonals of each square plaquette, the same diagonal in every square. Considered on an equivalent triangular-lattice geometry the model may be regarded as having two sorts of nearest-neighbor bonds, with $J_{2}' \equiv \kappa J_{1}$ bonds along parallel chains and $J_{1}$ bonds providing an interchain coupling. Hence, the model interpolates between a spin-half HAF on the square lattice at one extreme ($\kappa = 0$) and a set of decoupled spin-half chains at the other ($\kappa \to \infty$), with the spin-half HAF on the triangular lattice in between at $\kappa = 1$. We find strong evidence that quantum fluctuations favor a first-order transition from quasiclassical N\'{e}el order to a quantum helical state at a first critical point at $\kappa_{c_{1}} = 0.80 \pm 0.01$, by contrast with the corresponding second-order transition between the equivalent classical states at $\kappa_{{\rm cl}} = 0.5$. We also find strong evidence for a second critical point at $\kappa_{c_{2}} = 1.8 \pm 0.4$ where another first-order transition occurs, this time from the quantum helical phase to a collinear stripe-ordered phase. This latter result provides quantitative verification of a recent qualitative prediction of Starykh and Balents [Phys.\ Rev. Lett. {\bf 98}, 077205 (2007)] for the $J_{1}$--$J_{2}'$ model that did not, however, evaluate the corresponding critical point.