where are the Pauli matrices acting on the jth spin, hj determine the energy shifts of the spin-up and spin-down states playing the role of the local magnetic field, Jj are the nearest-neighbour spin–spin interactions, and Δ is the asymmetry parameter: Δ=0 corresponds to the purely spin-exchange XX model, Δ=1 to the homogeneous spin–spin interaction XXX model, while the limit of |Δ|>>1 leads to the Ising model. We assume a spatially symmetric spin chain with Jj=JN−j and hj=hN+1−j with h1,N=0. Note that we do not specify the sign of Jj, which, therefore, can be positive or negative.