  For the inverse square long-range ferromagnetic Ising chain in a transverse
field, the thermal phase boundary of the floating Kosterlitz-Thouless phase is
obtained for several values of the transverse field down to the quantum
critical point. The sharp domain walls in the classical model are increasingly
smeared out by the transverse field, which is evidenced by a pronounced
broadening of the non-universal bump in the specific heat. The discernability
of KT critical scaling in finite-size simulations is discussed.
