  We solve for the exact energy spectrum, 2-point and 4-point functions of the
complex SYK model, in the double scaling limit at all energy scales. This model
has a $U(1)$ global symmetry. The analysis shows how to incorporate a chemical
potential in the chord diagram picture, and we present results for the various
observables also at a given fixed charge sector. In addition to matching to the
spectral asymmetry, we consider an analogous asymmetry measure of the 2-point
function obeying a non-trivial dependence on the operator's dimension. We also
provide 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 exactly
compute the effect of inserting very heavy operators, with formally infinite
conformal dimension. The latter separate the gravitational spacetime into
several parts connected by an interface, whose properties are exactly
computable at all scales. In particular, light enough states can still go
between the spaces. This behavior has a simple description in the chord diagram
picture.
