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We study $\mathcal{N}=2$ supersymmetric Sachdev-Ye-Kitaev (SYK) models with
complex fermions at non-zero background charge. Motivated by multi-charge
supersymmetric black holes, we propose a new $\mathcal{N}=2$ SYK model with
multiple $U(1)$ symmetries, integer charges, and a non-vanishing supersymmetric
index, realizing features not present in known SYK models. In both models, a
conformal solution with a super-Schwarzian mode emerges at low temperatures,
signalling the appearance of nearly AdS$_2$/BPS physics. However, in contrast
to complex SYK, the fermion scaling dimension depends on the background charge
in the conformal limit. For a critical charge, we find a high to low entropy
phase transition in which the conformal solution ceases to be valid. This
transition has a simple interpretation: the fermion scaling dimension violates
the unitarity bound. We offer some comments on a holographic interpretation for
supersymmetric black holes.

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