Free semidefinite representation of matrix power functions

Consider the matrix power function Xp defined over the cone of positive definite matrices S++n. It is known that Xp is convex over S++n if p∈[−1,0]∪[1,2] and Xp is concave over S++n if p∈[0,1]. We show that the hypograph of Xp admits a free semidefinite representation if p∈[0,1] is rational, and the epigraph of Xp admits a free semidefinite representation if p∈[−1,0]∪[1,2] is rational.