  We study a model of spins 1/2 on a square lattice, generalizing the quantum
compass model via the addition of perturbing Heisenberg interactions between
nearest neighbors, and investigate its phase diagram and magnetic excitations.
This model has motivations both from the field of strongly correlated systems
with orbital degeneracy and from that of solid-state based devices proposed for
quantum computing. We find that the high degeneracy of ground states of the
compass model is fragile and changes into twofold degenerate ground states for
any finite amplitude of Heisenberg coupling. By computing the spin structure
factors of finite clusters with Lanczos diagonalization, we evidence a rich
variety of phases characterized by Z2 symmetry, that are either ferromagnetic,
C-type antiferromagnetic, or of Neel type, and analyze the effects of quantum
fluctuations on phase boundaries. In the ordered phases the anisotropy of
compass interactions leads to a finite excitation gap to spin waves. We show
that for small nanoscale clusters with large anisotropy gap the lowest
excitations are column-flip excitations that emerge due to Heisenberg
perturbations from the manifold of degenerate ground states of the compass
model. We derive an effective one-dimensional XYZ model which faithfully
reproduces the exact structure of these excited states and elucidates their
microscopic origin. The low energy column-flip or compass-type excitations are
robust against decoherence processes and are therefore well designed for
storing information in quantum computing. We also point out that the dipolar
interactions between nitrogen-vacancy centers forming a rectangular lattice in
a diamond matrix may permit a solid-state realization of the anisotropic
compass-Heisenberg model.
