= Solution
Using the consistently normalized <Jordan–Wigner transformation> from part (c), write $\sigma_1^x\sigma_2^x+\sigma_1^y\sigma_2^y=2(s_1^+s_2^-+s_1^-s_2^+)$. Since $f_1^\dagger P_1=f_1^\dagger$ and $f_1P_1=-f_1$, the parity string gives
$$
s_1^+s_2^-=f_1^\dagger f_2,\qquad s_1^-s_2^+=f_2^\dagger f_1.
$$
Therefore
$$
\boxed{H=2h-2h(n_1+n_2)-2J(f_1^\dagger f_2+f_2^\dagger f_1).}
$$
The <orthogonal transformation> $b=(f_1+f_2)/\sqrt2$, $c=(f_1-f_2)/\sqrt2$ preserves the <canonical anticommutation relations> and diagonalizes this bilinear <Hamiltonian>:
$$
\boxed{H=2h+(-2h-2J)b^\dagger b+(-2h+2J)c^\dagger c.}
$$
The <Fock vacuum> is $|\downarrow\downarrow\rangle$, with <energy> $2h$. Occupying $b$ alone gives the symmetric opposite-spin state with <energy> $-2J$; occupying $c$ alone gives the antisymmetric state with <energy> $2J$; occupying both gives $|\uparrow\uparrow\rangle$, up to a <global phase>, with <energy> $-2h$. This reproduces the entire spectrum in part (b).
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