The cluster state represents a highly entangled state which is one central
object for measurement-based quantum computing. Here we study the robustness of
the cluster state on the two-dimensional square lattice at zero temperature in
the presence of external magnetic fields by means of different types of
high-order series expansions and variational techniques using infinite
Projected Entangled Pair States (iPEPS). The phase diagram displays a
first-order phase transition line ending in two critical end points.
Furthermore, it contains a characteristic self-dual line in parameter space
allowing many precise statements. The self-duality is shown to exist on any
lattice topology.Comment: 12 pages, 9 figure
