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In the ordered product basis $(|\uparrow \uparrow ⟩,|\uparrow \downarrow ⟩,|\downarrow \uparrow ⟩,|\downarrow \downarrow ⟩)$, the two-spin XY Hamiltonian isThe transverse term exchanges opposite spins and leaves parallel spins unchanged. Diagonalizing its two-dimensional block gives the complete orthonormal eigenbasisThe antisymmetric state is the singlet state; the other three are triplet states. Accidental equality of energies does not invalidate this eigenbasis.

With occupied fermion sites representing up-spins, use $s_1^{-}=f_1$, $s_2^{-}=P_1{f}_2$ and $P_1=(-1{)}^{n_1}=1-2n_1$. The canonical anticommutation relations imply $P_1^2=I$, $P_1{f}_1=-f_1{P}_1$ and $[P_1,f_2]=0$. Consequently the two sites commute as spin operators, while $[s_j^{-},s_j^{+}]=1-2n_j=-{\sigma }_j^z$. This normalization is essential: the unscaled combinations ${\sigma }_j^x\pm i{\sigma }_j^y$ are twice $s_j^{\pm }$.

The two-spin XY Hamiltonian becomesIn particular, $f_1^{†}{P}_1=f_1^{†}$ and $f_1{P}_1=-f_1$ fix the hopping signs. The modes $b=(f_1+f_2)/\sqrt{2}$ and $c=(f_1-f_2)/\sqrt{2}$ preserve the canonical anticommutation relations and give $H=2h+(-2h-2J)b^{†}b+(-2h+2J)c^{†}c$.