  As shown in [1], two copies of the large $N$ Majorana SYK model can produce
spontaneous breaking of a $Z_2$ symmetry when they are coupled by appropriate
quartic terms. In this paper we similarly study two copies of the complex SYK
model coupled by a quartic term preserving the $U(1) \times U(1)$ symmetry. We
also present a tensor counterpart of this coupled model. When the coefficient
$\alpha$ of the quartic term lies in a certain range, the coupled large $N$
theory is nearly conformal. We calculate the scaling dimensions of fermion
bilinear operators as functions of $\alpha$. We show that the operator
$c_{1i}^\dagger c_{2i}$, which is charged under the axial $U(1)$ symmetry,
acquires a complex dimension outside of the line of fixed points. We derive the
large $N$ Dyson-Schwinger equations and show that, outside the fixed line, this
$U(1)$ symmetry is spontaneously broken at low temperatures because this
operator acquires an expectation value. We support these findings by exact
diagonalizations extrapolated to large $N$.
