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An antiperiodic boundary condition is a phase twist of $\pi$ across the boundary. A Fourier mode then has $k=2\pi (n+1/2)/L$. It occurs in the even-fermion sector of a periodic Jordan–Wigner transformation and in imaginary-time fermionic thermal fields, for different underlying reasons.

For $S_n^{+}=P_n{c}_n$ with $P_n={\prod }_{m<n}(1-2c_m^{†}{c}_m)$, neighbouring Jordan–Wigner transformation strings differ by only the local parity factor. The canonical anticommutation relations give $(c_n+c_n^{†})(1-2c_n^{†}{c}_n)=c_n^{†}-c_n$, proving the displayed local spin identity. The closing bond of a periodic chain instead retains total fermion parity; cancellation on bulk bonds does not justify discarding that global sector.

A bulk quasiparticle energy is not automatically the first excitation gap of a finite periodic spin chain. The Jordan–Wigner transformation fixes a boundary condition through fermion parity and restricts occupations to that same parity. At zero transverse field an Ising domain wall costs 2J, but periodic spin configurations contain an even number of walls; the first wall-pair excitation costs 4J. An ordered finite chain also has a distinct small symmetry-sector splitting at nonzero field. The thermodynamic dispersion and these finite-system scales answer different questions.

For exchange $J>0$, the Jordan–Wigner transformation gives a band ${ϵ}_k=h-J\cos k$. At zero temperature, negative-energy states form a filled interval $|k|<\arccos (h/J)$. Since each fermion decreases total magnetization by one, the thermodynamic magnetization per site is $1/2-\arccos (h/J)/\pi$. It saturates at $1/2$ for $h\ge J$. A finite ring instead has discrete level crossings and parity-dependent momentum grids.

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.

Let $n_n=c_n^{†}{c}_n$ and $P_n={\prod }_{m<n}(1-2n_m)$. The Jordan–Wigner transformation gives $2S_n^x=P_n(c_n+c_n^{†})$ and $P_{n+1}=P_n(1-2n_n)$. The canonical anticommutation relations implyConsequently the bondwise cancellation of a Jordan-Wigner string yieldsSubstitution therefore givesThe finite-ring parity qualification of an Ising quasiparticle gap matters here. For an exact periodic spin chain, the closing bond retains total fermion parity. The parity-dependent boundary condition of the Jordan-Wigner chain is $c_{N+1}=-P{c}_1$, with $P=(-1{)}^{N_f}$. Fixing that parity determines the allowed Fourier grid, and the resulting many-body states must still have the chosen parity. Dropping this boundary issue preserves the thermodynamic energy density, but can lose a degeneracy or the correct finite-ring excitation gap. ^[ref]

Use the canonical anticommutation relations $\{c_n,c_m^{†}\}={\delta }_{nm}$ and $\{c_n,c_m\}=0$. In particular, $n_m=c_m^{†}{c}_m$ is a projector, so $(1-2n_m{)}^2=1$. Define $K_n={\prod }_{m<n}(1-2n_m)$. Then $K_n^{†}=K_n$, $K_n^2=1$, and $K_n$ commutes with $c_n$ and $c_n^{†}$. Taking the adjoint of the given raising operator gives $S_n^{-}=c_n^{†}{K}_n$. HenceAlso $[n_n,c_n]=-c_n$ and $[n_n,c_n^{†}]=c_n^{†}$ imply $[S_n^z,S_n^{\pm }]=\pm S_n^{\pm }$. For different sites, exchanging two fermion operators contributes one minus sign and commuting an operator through the other site's parity string contributes another; their product is positive. Thus distinct spin algebras commute. The string supplies precisely the signs required for the Jordan–Wigner transformation. The fermion vacuum corresponds to spin up, while one occupied site corresponds to spin down; this convention fixes subsequent signs.

There is a normalization mismatch in the printed definitions: ${\sigma }^x\pm i{\sigma }^y$ have nonzero entries of magnitude two, whereas a single fermionic creation or annihilation operator has magnitude one. To obtain the intended Jordan–Wigner transformation, use normalized spin ladders $s_j^{\pm }=({\sigma }_j^x\pm i{\sigma }_j^y)/2$. The given fermion expressions represent these $s_j^{\pm }$. The two-site Jordan-Wigner normalization and signs then fix every factor unambiguously.

Put $n_j=f_j^{†}{f}_j$ and $P_1=e^{i\pi n_1}=1-2n_1$. The canonical anticommutation relations give $P_1^2=I$, $P_1{f}_1=-f_1{P}_1$, and $[P_1,f_2]=[P_1,f_2^{†}]=0$. HenceandThese are the spin commutation relations, with distinct sites commuting. If one retains the unscaled symbols ${\sigma }_j^{\pm }=2s_j^{\pm }$, the first and last displayed commutators instead equal $-4{\sigma }_j^z$; the off-site commutators remain zero. Setting the unscaled ${\sigma }_j^{-}$ equal to $f_j$ would contradict the original Pauli matrices.

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 $f_1^{†}{P}_1=f_1^{†}$ and $f_1{P}_1=-f_1$, the parity string givesThereforeThe orthogonal transformation $b=(f_1+f_2)/\sqrt{2}$, $c=(f_1-f_2)/\sqrt{2}$ preserves the canonical anticommutation relations and diagonalizes this bilinear Hamiltonian:The Fock vacuum is $|\downarrow \downarrow ⟩$, 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 $|\uparrow \uparrow ⟩$, up to a global phase, with energy $-2h$. This reproduces the entire spectrum in part (b).

The spin-one-half quantum XX chain has equal nearest-neighbour transverse exchange. Its continuous spin rotation symmetry conserves total longitudinal magnetization. The Jordan–Wigner transformation maps it to number-conserving free fermions. This quantum chain is distinct from the classical XY model.