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As shown in [1], two copies of the large $N$ Majorana SYK model can producespontaneous breaking of a $Z_2$ symmetry when they are coupled by appropriatequartic terms. In this paper we similarly study two copies of the complex SYKmodel coupled by a quartic term preserving the $U(1) \times U(1)$ symmetry. Wealso 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 fermionbilinear 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 thelarge $N$ Dyson-Schwinger equations and show that, outside the fixed line, this$U(1)$ symmetry is spontaneously broken at low temperatures because thisoperator acquires an expectation value. We support these findings by exactdiagonalizations extrapolated to large $N$.

Spontaneous Breaking of $U(1)$ Symmetry in Coupled Complex SYK Models