I am reading Coleman's "Introduction to many-body physics" and am working on problem 4.2, which involves calculating the spectrum of the transverse-field Ising model. We start with the Hamiltonian H=−J∑jSjxSj+1x−h∑jSjz.H = -J \sum_{j} S^{x}_{j} S^{x}_{j+1} -h \sum_{j} S^{z}_{j}. I have successfully used a Jordan-Wigner transformation to turn it into fermion creation/annihilation operators as H=∑j(J4(fjfj+1+fjfj+1†−fj†fj+1−fj†fj+1†)−hfj†fj). H = \sum_j \left( \frac{J}{4}\left(f_jf_{j+1}+f_jf^{\dagger}_{j+1}-f_j^{\dagger}f_{j+1}-f_j^{\dagger}f_{j+1}^{\dagger} \right)-hf_j^{\dagger}f_j\right). Then I Fourier transformed this Hamiltonian to get H=∑k(Adk†dk−Jisin⁡ka4(dkd−k+dk†d−k†)),H = \sum_k \left(Ad_k^{\dagger}d_k -\frac{Ji\sin{ka}}{4}\left(d_kd_{-k}+d_k^{\dagger}d_{-k}^{\dagger}\right) \right), where A=−J2cos⁡ka−hA = -\frac{J}{2}\cos{ka}-h. The textbook says the sum should be only over half of the Brillouin zone k∈[0,π/a]k\in[0,\pi/a], but I don't understand why.