Two-spin XY Hamiltonian ($H=-J({\sigma }_1^x{\sigma }_2^x+{\sigma }_1^y{\sigma }_2^y)-h({\sigma }_1^z+{\sigma }_2^z)$)

The two-site quantum version of an XY model couples the transverse components of two spin one-half degrees of freedom. In the ordered basis $|\uparrow \uparrow ⟩,|\uparrow \downarrow ⟩,|\downarrow \uparrow ⟩,|\downarrow \downarrow ⟩$, its only off-diagonal entries are $H_{23}=H_{32}=-2J$. The parallel states have energies $-2h$ and $2h$; the symmetric and antisymmetric opposite-spin combinations have energies $-2J$ and $2J$ respectively. The latter is the singlet state. The formulas remain valid at accidental degeneracies.