Schwinger model

In physics, the Schwinger model, named after Julian Schwinger, is the model describing 2D (1 spatial dimension + time) Euclidean quantum electrodynamics with a Dirac fermion. This model exhibits a spontaneous symmetry breaking of the U(1) symmetry due to a chiral condensate due to a pool of instantons. The photon in this model becomes a massive particle at low temperatures. This model can be solved exactly and is used as a toy model for other more complex theories.

This model exhibits confinement of the fermions and as such, is a toy model for QCD. A handwaving argument why this is so is because in two dimensions, classically, the potential between two charged particles goes linearly as $r$, instead of $1/r$ in 4 dimensions (3 spatial 1 time).