  We prove that when suitably normalized, small enough powers of the absolute
value of the characteristic polynomial of random Hermitian matrices, drawn from
one-cut regular unitary invariant ensembles, converge in law to Gaussian
multiplicative chaos measures. We prove this in the so-called $L^2$-phase of
multiplicative chaos. Our main tools are asymptotics of Hankel determinants
with Fisher-Hartwig singularities. Using Riemann-Hilbert methods, we prove a
rather general Fisher-Hartwig formula for one-cut regular unitary invariant
ensembles.
