  In this paper, we study the smallest non-zero eigenvalue of the sample
covariance matrices $\mathcal{S}(Y)=YY^*$, where $Y=(y_{ij})$ is an $M\times N$
matrix with iid mean $0$ variance $N^{-1}$ entries. We prove a phase transition
for its distribution, induced by the fatness of the tail of $y_{ij}$'s. More
specifically, we assume that $y_{ij}$ is symmetrically distributed with tail
probability $\mathbb{P}(|\sqrt{N}y_{ij}|\geq x)\sim x^{-\alpha}$ when $x\to
\infty$, for some $\alpha\in (2,4)$. We show the following conclusions: (i).
When $\alpha>\frac83$, the smallest eigenvalue follows the Tracy-Widom law on
scale $N^{-\frac23}$; (ii). When $2<\alpha<\frac83$, the smallest eigenvalue
follows the Gaussian law on scale $N^{-\frac{\alpha}{4}}$; (iii). When
$\alpha=\frac83$, the distribution is given by an interpolation between
Tracy-Widom and Gaussian; (iv). In case $\alpha\leq \frac{10}{3}$, in addition
to the left edge of the MP law, a deterministic shift of order
$N^{1-\frac{\alpha}{2}}$ shall be subtracted from the smallest eigenvalue, in
both the Tracy-Widom law and the Gaussian law. Overall speaking, our proof
strategy is inspired by \cite{ALY} which is originally done for the bulk regime
of the L\'{e}vy Wigner matrices. In addition to various technical complications
arising from the bulk-to-edge extension, two ingredients are needed for our
derivation: an intermediate left edge local law based on a simple but effective
matrix minor argument, and a mesoscopic CLT for the linear spectral statistic
with asymptotic expansion for its expectation.
