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If the traceless matrix $h_k$ has eigenvalues $\pm d_k$ and the Nambu spinor includes an opposite-momentum creation operator, a full-zone sum contains both partners. Its physical quasiparticle energy is $2J{d}_k$, not $J{d}_k$. The negative term becomes a vacuum constant only after normal ordering. In the Ising-chain Nambu spinor, $d_k=\sqrt{1+g^2-2g\cos k}$. A half-zone calculation instead groups one complete two-mode block; changing the momentum range without changing the normalization is inconsistent.

For a paired non-self-conjugate momentum, rotate the Ising-chain Nambu spinor by an angle with $\cos \theta =(g-\cos k)/r_k$, $\sin \theta =-\sin k/r_k$ and $r_k=\sqrt{1+g^2-2g\cos k}$. Opposite angles at $k,-k$ preserve the canonical anticommutation relations. The result is $H={\sum }_k{ϵ}_k({\gamma }_k^{†}{\gamma }_k-1/2)$. Self-paired momenta have no pairing and are handled directly as number levels. Periodic spin-chain sectors additionally constrain fermion parity and allowed mode quantization.

Put $n_n=c_n^{†}{c}_n$ and $P_n={\prod }_{m<n}(1-2n_m)$. The Jordan–Wigner transformation is $S_n^{+}=P_n{c}_n$, $S_n^{-}=P_n{c}_n^{†}$ and $S_n^z=1/2-n_n$. Since ${\sigma }_n^x=P_n(c_n+c_n^{†})$, $P_{n+1}=P_n(1-2n_n)$ and $(c_n+c_n^{†})(1-2n_n)=c_n^{†}-c_n$, adjacent bonds becomeThus, apart from the end bond,The end bond contains the global fermion parity and sets the sector-dependent periodic or antiperiodic fermion modes. It contributes an order-one boundary term, which is negligible for the thermodynamic energy density; it is not identically zero for the finite spin chain.

Choose the discrete Fourier transform convention $c_n=N^{-1/2}{\sum }_k{e}^{-ikn}{c}_k$. Hopping gives $-2J\cos k{c}_k^{†}{c}_k$. Opposite-momentum pairing gives $iJ\sin k(c_k^{†}{c}_{-k}^{†}+c_k{c}_{-k})$, using the canonical anticommutation relations to antisymmetrize the coefficient. ThereforeFor the Ising-chain Nambu spinor ${\Psi }_k=(c_k,-i{c}_{-k}^{†}{)}^T$, expansion ofrecovers every term: the diagonal contributes $J(g-\cos k)(n_k+n_{-k}-1)$, and the two off-diagonal entries supply the pairing. Since ${\sum }_k\cos k=0$, the constant is $-JgN$. Reversing the Fourier sign changes the pairing convention; the specified sign makes the displayed matrix agree directly.

With the specified Fourier transform convention, normal ordering giveswhere the possible constant ${\sum }_k{e}^{ik}$ vanishes on the complete lattice momentum grid. Antisymmetry of opposite-momentum pairs similarly givesHenceUse the Ising-chain Nambu spinor ${\Psi }_k=(c_k,-i{c}_{-k}^{†}{)}^T$. Expanding its two diagonal terms uses $c_{-k}{c}_{-k}^{†}=1-c_{-k}^{†}{c}_{-k}$, retaining the constant. The result isThis is a full-zone sum. Its opposite-momentum components are redundant; treating each matrix eigenvalue as an independent physical excitation would miss the factor of two in the quasiparticle energy. Self-paired momenta have zero anomalous term and are ordinary number levels.

Put $d_k=\sqrt{(g-\cos k{)}^2+{\sin }^{2}k}$. The matrix in the Ising-chain Nambu spinor has eigenvalues $\pm d_k$. A canonical Bogoliubov transformation, with paired angles satisfying $\cos {\vartheta }_k=(g-\cos k)/d_k$ and $\sin {\vartheta }_k=-\sin k/d_k$, diagonalizes it. Accounting for full-zone fermionic Nambu energy counting givesThus the bulk Ising-chain ground-state energy isExact finite-chain states additionally obey their fermion parity sector.

The Ising-chain quasiparticle dispersion has bulk gap $2J|1-g|$, closing at the Ising-chain quantum critical point. Finite-system parity splittings and the bulk gap are distinct scales. The three requested limits are illustrated below.