Even-odd effects in the Ising and XY chains.

Boundary conditions were traditionally believed not to strongly influence the properties of condensed-matter systems. This belief has been proven wrong with the advent of topological insulators, where systems with gapped bulk bands are bound to have metallic states at their physical terminations. An even more striking violation of the paradigm has recently been discovered in some antiferromagnetic spin chains with competing orders. What has been shown in this context is that, depending on the parity of the number of spins, these systems can either be gapless or gapped, and that different order parameters need to be introduced. In my contribution, I first report about a first-order quantum phase transition that is present in the quantum Ising model in a transverse field for an odd number of spins only. Subsequently, I report on how, in the XY chain, the parity of the number of spins can alter the order of the quantum phase transitions of the model. Finally, I describe how, again in the XY model, a full extra critical line appears in the odd spin case only.