  We show that the scaling dimensions of lowest operators in conformal field
theories (CFTs) can be isolated in small and closed regions from single
correlator bootstrap. We find the conserved currents play crucial roles in
bootstrapping the crossing equation. By imposing a mild gap between the scaling
dimensions of the conserved current and its next operator, the scaling
dimensions of lowest operators are forced to lie in small isolated regions,
i.e., these CFTs can be almost fixed by few lowest operators in certain
channels. For CFTs with extended supersymmetry, the single correlator crossing
equation involves several conserved or shorted operators and by imposing gaps
in these sectors it is possible to isolate different CFTs. Specifically, we
bootstrap the isolated regions corresponding to the 3D Ising model, $O(N)$
vector model, $N=1,2$ supersymmetric Ising models by introducing mild gaps in
certain sectors with conserved or shorted operators.
