Parity-Dependent Quantum Phase Transition in the Quantum Ising Chain in a Transverse Field

Phase transitions-both classical and quantum types-are the perfect playground for appreciating universality at work. Indeed, the fine details become unimportant and a classification in very few universality classes is possible. Very recently, a striking deviation from this picture has been discovered: some antiferromagnetic spin chains with competing interactions show a different set of phase transitions depending on the parity of number of spins in the chain. The aim of this article is to demonstrate that the same behavior also characterizes the most simple quantum spin chain: the Ising model in a transverse field. By means of an exact solution based on a Wigner-Jordan transformation, we show that a first-order quantum phase transition appears at the zero applied field in the odd spin case, while it is not present in the even case. A hint of a possible physical interpretation is given by the combination of two facts: at the point of the phase transition, the degeneracy of the ground state in the even and the odd case substantially differs, being respectively 2 and 2N, with N being the number of spins; the spin of the most favorable kink shows changes at that point.