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We solve for the exact energy spectrum, 2-point and 4-point functions of thecomplex SYK model, in the double scaling limit at all energy scales. This modelhas a $U(1)$ global symmetry. The analysis shows how to incorporate a chemicalpotential in the chord diagram picture, and we present results for the variousobservables also at a given fixed charge sector. In addition to matching to thespectral asymmetry, we consider an analogous asymmetry measure of the 2-pointfunction obeying a non-trivial dependence on the operator's dimension. We alsoprovide the chord diagram structure for an SYK-like model that has a $U(M)$global symmetry at any disorder realization. We then show how to exactlycompute the effect of inserting very heavy operators, with formally infiniteconformal dimension. The latter separate the gravitational spacetime intoseveral parts connected by an interface, whose properties are exactlycomputable at all scales. In particular, light enough states can still gobetween the spaces. This behavior has a simple description in the chord diagrampicture.

Complex Sachdev-Ye-Kitaev model in the double scaling limit