GE in other 2D quantum models.

2D spin-$\frac12$ XXX model in a magnetic field
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The spin-$\frac12$ antiferromagnetic XXX model on the square lattice has hamiltonian $$H =
  \sum_{( \vec r, \vec r')} \left( S^{[\vec r]}_x S^{[\vec r']}_x
    +  S^{[\vec r]}_y S^{[\vec r']}_y
    +  S^{[\vec r]}_z S^{[\vec r']}_z \right)-h \sum_{\vec r} S^{[\vec r]}_z,$$ where $h$ is an external magnetic field along the $z$ direction. This model has been studied via a barrage of different techniques [@sandvik]. Whereas magnetic order is normally ruled out in 1D Heisenberg models, this is not the case for 2D Heisenberg models [@mermin; @sandvik]. Thus in 2D the ground state can be non-magnetic, i.e., with a non-zero magnetization. In the absence of the magnetic field the ground state of the Heisenberg model has antiferromagnetic (Néel) order with a non-zero local staggered magnetization and infinitely degenerate ground states resulting from the breaking of global SU$(2)$ spin rotation symmetry. For $h\neq0$, infinitely degenerate ground states are detected resulting from the spontaneous symmetry breaking of U$(1)$ symmetry in the $x$-$y$ plane. As $h\rightarrow \infty$ it is anticipated the system will be fully polarized in the $z$ direction. In fact this transition to the fully polarized state occurs at $h=4$.

