  We consider the highly spin-imbalanced limit of a two-component Fermi gas,
where there is a small density of $\downarrow$ impurities attractively
interacting with a sea of $\uparrow$ fermions. In the single-impurity limit at
zero temperature, there exists the so-called polaron-molecule transition, where
the impurity sharply changes its character by binding a $\uparrow$ fermion at
sufficiently strong attraction. Using a recently developed variational
approach, we calculate the thermodynamic properties of the impurity, and we
show that the transition becomes a smooth crossover at finite temperature due
to the thermal occupation of excited states in the impurity spectral function.
However, remnants of the single-impurity transition are apparent in the
momentum-resolved spectral function, which can in principle be probed with
Raman spectroscopy. We furthermore show that the Tan contact exhibits a
characteristic non-monotonic dependence on temperature that provides a
signature of the zero-temperature polaron-molecule transition. For a finite
impurity density, we argue that descriptions purely based on the behavior of
the Fermi polaron are invalid near the polaron-molecule transition, since
correlations between impurities cannot be ignored. In particular, we show that
the spin-imbalanced system undergoes phase separation at low temperatures due
to the strong attraction between $\uparrow\downarrow$ molecules induced by the
Fermi sea. Thus, we find that the impurity spectrum and the induced
impurity-impurity interactions are key to understanding the phase diagram of
the spin-imbalanced Fermi gas.
