(b) Amplitude of the local order parameters $M$. (c) GE per site ${\mathcal E}(h)$. []{data-label="fig3"}](delta10c.eps "fig:"){width="0.8\linewidth"}

2D spin-$\frac12$ XYX model in a magnetic field
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Tuning the anisotropy of the Heisenberg interactions leads to the more general spin-$\frac12$ XYX antiferromagnetic model in a uniform magnetic field, defined on the square lattice by the hamiltonian $$H =
  \sum_{( \vec r, \vec r')} \left( S^{[\vec r]}_x S^{[\vec r']}_x
    +\Delta_{y}  S^{[\vec r]}_y S^{[\vec r']}_y
    +S^{[\vec r]}_z S^{[\vec r']}_z \right)
    -h \sum_{\vec r} S^{[\vec r]}_z.$$ Varying the anisotropic exchange interaction parameter $\Delta_y$ leads to different behavior, with the two cases $\Delta_{y}<1$ and $\Delta_{y}>1$ corresponding to an easy-plane and easy-axis behavior, respectively. The quantum criticality of this model is well understood [@tros], with an ordered phase below a critical field value $h_c$, above which is a partially polarized state with field-induced magnetization reaching saturation as $h \to \infty$. The ordered phase in the easy-plane (easy-axis) case arises by spontaneous symmetry breaking along the $x$ ($y$) direction, which corresponds to a finite value of the order parameter $M_x$ ($M_y$) below $h_c$. At the transition point $h_c$, long-range correlations are destroyed.

Previous studies of this model using GE [@gehlw; @hl] focussed on the value $\Delta_{y}=0.25$ with the 