Distinguishing topologically and smoothly doubly slice knots

We construct an infinite family of smoothly slice knots that we prove are topologically doubly slice. Using the correction terms coming from Heegaard Floer homology, we show that none of these knots is smoothly doubly slice. We use these knots to show that the subgroup of the double concordance group consisting of smoothly slice, topologically doubly slice knots is infinitely generated. As a corollary, we produce an infinite collection of rational homology 3-spheres that embed in $S^4$ topologically, but not smoothly.