One-dimensional physics

There are quite a few special aspects of many-body physics in one dimension. One of the most important is that the difference between bosons (including spins, as spin operators commute on different sites, like Bose operators) and fermions, which is dramatic in higher dimensions, is surprisingly small in one dimension. We give an example of equivalence between fermonic and bosonic particles via the Jordan-Wigner transform applied to the "XX model" of spins, which turns out to be free fermions in disguise. The basics of the Bethe ansatz solution of certain one-dimensional interacting models are also described.