Magnetic order on a frustrated spin- 12 Heisenberg antiferromagnet on the Union Jack lattice

Abstract

We use the coupled cluster method (CCM) to study the zero-temperature phase diagram of a two-dimensional frustrated spin-half antiferromagnet, the so-called Union Jack model. It is defined on a square lattice such that all nearest-neighbor pairs are connected by bonds with a strength ${J}_{1}>0$, but only half the next-nearest-neighbor pairs are connected by bonds with a strength ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$. The bonds are arranged such that on the $2\ifmmode\times\else\texttimes\fi{}2$ unit cell they form the pattern of the Union Jack flag. Alternating sites on the square lattice are thus four-connected and eight-connected. We find strong evidence for a first phase transition between a N\'eel antiferromagnetic phase and a canted ferrimagnetic phase at a critical coupling ${\ensuremath{\kappa}}_{{c}_{1}}=0.66\ifmmode\pm\else\textpm\fi{}0.02$. The transition is an interesting one, at which the energy and its first derivative seem continuous, thus providing a typical scenario of a second-order transition (just as in the classical case for the model), although a weakly first-order transition cannot be excluded. By contrast, the average on-site magnetization approaches a nonzero value ${M}_{{c}_{1}}=0.195\ifmmode\pm\else\textpm\fi{}0.005$ on both sides of the transition, which is more typical of a first-order transition. The slope, $dM/d\ensuremath{\kappa}$, of the order parameter curve as a function of the coupling strength $\ensuremath{\kappa}$, also appears to be continuous, or very nearly so, at the critical point ${\ensuremath{\kappa}}_{{c}_{1}}$, thereby providing further evidence of the subtle nature of the transition between the N\'eel and canted phases. Our CCM calculations provide strong evidence that the canted ferrimagnetic phase becomes unstable at large values of $\ensuremath{\kappa}$, and hence we have also used the CCM with a model collinear semistripe-ordered ferrimagnetic state in which alternating rows (and columns) are ferromagnetically and antiferromagnetically ordered, and in which the spins connected by ${J}_{2}$ bonds are antiparallel to one another. We find tentative evidence, based on the relative energies of the two states, for a second zero-temperature phase transition between the canted and semistripe-ordered ferrimagnetic states at a large value of the coupling parameter around ${\ensuremath{\kappa}}_{{c}_{2}}\ensuremath{\approx}125\ifmmode\pm\else\textpm\fi{}5$. This prediction, however, is based on an extrapolation of the CCM results for the canted state into regimes where the solutions have already become unstable and the CCM equations based on the canted state at any level of approximation beyond the lowest have no solutions. Our prediction for ${\ensuremath{\kappa}}_{{c}_{2}}$ is hence less reliable than that for ${\ensuremath{\kappa}}_{{c}_{1}}$. Nevertheless, if this second transition at ${\ensuremath{\kappa}}_{{c}_{2}}$ does exist, our results clearly indicate it to be of first-order type.