  We study the SU(3) gauge theory with Nf=12 flavors in the fundamental
representation by use of lattice simulations with staggered fermions. With a
non-improved action we observe a chiral zero-temperature (bulk) transition
separating a region at weak coupling, where chiral symmetry is realized, from a
region at strong coupling where chiral symmetry is broken. With improved
actions, a more complicated pattern emerges, and in particular two first order
transitions in the chiral limit appear. We observe that at sufficiently strong
coupling the next-to-nearest neighbor terms of the improved lattice action are
no longer irrelevant and can indeed modify the pattern observed without
improvement. Baryon number conservation can be realized in an unusual way,
allowing for an otherwise prohibited oscillating term in the pseudoscalar
channel. We discuss the phenomenon by means of explicit examples borrowed from
statistical mechanics. Finally, these observations can also be useful when
simulating other strongly coupled systems on the lattice, such as graphene.
