  We investigate a class of four-fermion theories which includes well-known
models like the Gross-Neveu model and the Thirring model. In three spacetime
dimensions, they are used to model interesting solid state systems like high
temperature superconductors and graphene. Additionally, they serve as toy
models to study chiral symmetry breaking (CSB).
  For any number of fermion flavours the Gross-Neveu model has a broken and a
symmetric phase, while the existence of a broken phase in the Thirring model
depends on the number of flavours. The critical number of fermion flavours
beyond which there exists no CSB is still subject of ongoing discussions. Using
SLAC fermions we simulate the Thirring model with exact chiral symmetry. To
obtain a chiral condensate one can introduce a symmetry-breaking mass term and
carefully study the limits of infinite lattice and zero-mass. So far, we did
not see CSB within this approach for the Thirring model with 2 or more
(reducible) flavours.
  The talk presents alternative approaches to investigate these findings. We
employ certain Fierz identities to map the Thirring model into equivalent
four-fermion models, for which the chiral condensate does not seem to vanish.
In the new formulations based on reshuffled degrees of freedom we find a sign
problem (which is not present in the original formulation). For this reason we
developed an algorithm similar to fermion bags, which may solve this problem.
As a further approach, we embed the multi-flavour Thirring model in a larger
class of four-fermion theories to study the chiral symmetry and its breaking in
a wider context.
