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We consider the square lattice $S$=1/2 quantum compass model (QCM)parameterized by $J_x, J_z$, under a field, $\mathbf{h}$, in the $x$-$z$ plane.At the special field value, $(h_x^\star,h_z^\star)$=$2S(J_x,J_z)$, we show thatthe QCM Hamiltonian may be written in a form such that two simple productstates can be identified as exact ground-states, below a gap. Exact excitedstates can also be found. The exact product states are characterized by astaggered vector chirality, attaining a non-zero value in the surroundingphase. The resulting gapped phase, which we denote by $SVC$ occupies most ofthe in-plane field phase diagram. For some values of $h_x>h_z$ and $h_z>h_x$ atthe edges of the phase diagram, we have found transitions between the $SVC$phase and phases of weakly-coupled Ising-chain states, $Z$ and $X$. In zerofield, the QCM is known to have an emergent sub-extensive ground-statedegeneracy. As the field is increased from zero, we find that this degeneracyis partially lifted, resulting in bond-oriented spin-stripe states, $L$ and$R$, which are each separated from one another and the $SVC$ phase byfirst-order transitions. Our findings are important for understanding the fielddependent phase diagram of materials with predominantly directionally-dependentIsing interactions.

Exact Ground States and Phase Diagram of the Quantum Compass Model under an in-plane Field