  We study U(1) gauge theory on a 4d non-commutative torus, where two
directions are non-commutative. Monte Carlo simulations are performed after
mapping the regularized theory onto a U(N) lattice gauge theory in d=2. At
intermediate coupling strength, we find a phase in which open Wilson lines
acquire non-zero vacuum expectation values, which implies the spontaneous
breakdown of translational invariance. In this phase, various physical
quantities obey clear scaling behaviors in the continuum limit with a fixed
non-commutativity parameter theta, which provides evidence for a possible
continuum theory. In the weak coupling symmetric phase, the dispersion relation
involves a negative IR-singular term, which is responsible for the observed
phase transition.
