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 $J_x,J_z$, under a field, $h$, in the $x\text{−}z$ plane. At the special field value $(h_x^{★},h_z^{★})=2S(J_x,J_z)$, 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 $(h_x^{★},h_z^{★})$ and states obtained for fields within the vicinity of $(h_x^{★},h_z^{★})$. This gapped phase occupies most of the in-plane field phase diagram. For some values of $h_x>h_z$ and $h_z>h_x$ 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.