  The Thirring model is an interacting fermion theory with current-current
interaction. The model in $1+2$ dimensions has applications in condensed-matter
physics to describe the electronic excitations of Dirac materials. Earlier
investigations with Schwinger-Dyson equations, the functional renormalization
group and lattice simulations with staggered fermions suggest that a critical
number of (reducible) flavors $N^{\mathrm{c}}$ exists, below which chiral
symmetry can be broken spontaneously. Values for $N^{\mathrm{c}}$ found in the
literature vary between $2$ and $7$. Recent lattice studies with chirally
invariant SLAC fermions have indicated that chiral symmetry is unbroken for all
integer flavor numbers [Wellegehausen et al., 2017]. An independent simulation
based on domain wall fermions seems to favor a critical flavor-number that
satisfies $1<N^{\mathrm{c}}<2$ [Hands, 2018]. However, in the latter
simulations difficulties in reaching the massless limit in the broken phase (at
strong coupling and after the $L_s\to\infty$ limit has been taken) are
encountered. To find an accurate value $N^{\mathrm{c}}$ we study the Thirring
model (by using an analytic continuation of the parity even theory to arbitrary
real $N$) for $N$ between $0.5$ and $1.1$. We investigate the chiral
condensate, the spectral density of the Dirac operator, the spectrum of
(would-be) Goldstone bosons and the variation of the filling-factor and
conclude that the critical flavor number is $N^{\mathrm{c}}=0.80(4)$. Thus we
see no chiral symmetry breaking in all Thirring models with $1$ or more flavors
of ($4$-component) fermions. Besides the artifact transition to the unphysical
lattice artifact phase we find strong evidence for a hitherto unknown phase
transition that exists for $N>N^{\mathrm{c}}$ and should answer the question of
where to construct a continuum limit.
