A variety of lattice discretisations of continuum actions has
been considered, by default with the correct classical continuum
limit. Here we discuss "weird" lattice formulation without that
property, namely lattice actions that are invariant under small
deformations of the field configuration, in some cases without
any coupling constants.
It turns out that universality is powerful enough to still
provide the correct quantum continuum limit, despite the absence
of any classical limit, or a perturbative expansion. We demonstrate
this for a set of O(N) models, including the 1d case, where
universality is not even expected.
In addition to this conceptual insight, such "weird" lattice
actions even have amazing practical benefits, in particular
an excellent scaling behavior. In the 2d Heisenberg model this
study further clarifies the fate of the topological susceptibility.
In the 2d XY model we obtain evidence for an unexplored phase
transition, and for an unconventional understanding of the
vortex-antivortex binding mechanism.