  The two-species Fermi gas with attractive short-range interactions in two
spatial dimensions provides a paradigmatic system for the understanding of
strongly correlated Fermi superfluids in two dimensions. It is known to exhibit
a BEC-BCS crossover as a function of $\ln(k_F a)$, where $a$ is the scattering
length, and to undergo a Berezinskii-Kosterlitz-Thouless superfluid transition
below a critical temperature $T_c$. However, the extent of a pseudogap regime
in the strongly correlated regime of $\ln(k_F a)\sim 1$, in which pairing
correlations persist above $T_c$, remains largely unexplored with controlled
theoretical methods. Here we use finite-temperature auxiliary-field quantum
Monte Carlo (AFMC) methods on discrete lattices in the canonical ensemble
formalism to calculate thermodynamical observables in the strongly correlated
regime. We extrapolate to continuous time and the continuum limit to eliminate
systematic errors and present results for particle numbers ranging from $N=42$
to $N=162$. We estimate $T_c$ by a finite-size scaling analysis, and observe
clear pseudogap signatures above $T_c$ and below a temperature $T^*$ in both
the spin susceptibility and free-energy gap. We also present results for the
contact, a fundamental thermodynamic property of quantum many-body systems with
short-range interactions.
