Exact ground states and phase diagram of the quantum compass model under an in-plane field

We consider the square lattice $S=1/2$ quantum compass model (QCM) parametrized by Jₓ,Jz, under a field, h, in the x-z plane. At the special field value (hₓ★,hz★)=2S(Jₓ,Jz), we show that the QCM Hamiltonian may be written in a form such that two simple product states can be identified as exact ground states, below a gap. Exact excited states can also be found. We use the staggered vector chirality to characterize the exact ground states at (hₓ★,hz★) and states obtained for fields within the vicinity of (hₓ★,hz★). This gapped phase occupies most of the in-plane field phase diagram. For some values of hₓ>hz and hz>hₓ at the edges of the phase diagram, we have found transitions between this gapped phase and phases of weakly coupled Ising-chain states. In zero field, the QCM is known to have an emergent subextensive ground-state degeneracy. As the field is increased from zero, we find that this degeneracy is partially lifted, resulting in bond-oriented spin-stripe states, which are each separated from one another and the gapped phase by first-order transitions. Our findings are important for understanding the field dependent phase diagram of materials with predominantly directionally dependent Ising interactions.