  The Thirring model is a four-fermion theory with a current-current
interaction and $U(2N)$ chiral symmetry. It is closely related to
three-dimensional QED and other models used to describe properties of graphene.
In addition it serves as a toy model to study chiral symmetry breaking. In the
limit of flavour number $N \to 1/2$ it is equivalent to the Gross-Neveu model,
which shows a parity-breaking discrete phase transition. The model was already
studied with different methods, including Dyson-Schwinger equations, functional
renormalisation group methods and lattice simulations. Most studies agree that
there is a phase transition from a symmetric phase to a spontaneously broken
phase for a small number of fermion flavours, but no symmetry breaking for
large $N$. But there is no consensus on the critical flavour number
$N^\text{cr}$ above which there is no phase transition anymore and on further
details of the critical behaviour. Values of $N$ found in the literature vary
between $2$ and $7$.
  All earlier lattice studies were performed with staggered fermions. Thus it
is questionable if in the continuum limit the lattice model recovers the
internal symmetries of the continuum model. We present new results from lattice
Monte Carlo simulations of the Thirring model with SLAC fermions which exactly
implement all internal symmetries of the continuum model even at finite lattice
spacing. If we reformulate the model in an irreducible representation of the
Clifford algebra, we find, in contradiction to earlier results, that the
behaviour for even and odd flavour numbers is very different: For even flavour
numbers, chiral and parity symmetry are always unbroken. For odd flavour
numbers parity symmetry is spontaneously broken below the critical flavour
number $N_\text{ir}^\text{cr}=9$ while chiral symmetry is still unbroken.
