  The underlying physics of the magnetic-field-induced resistive state in high
temperature cuprate superconductors remains a mystery. One interpretation is
that the application of magnetic field destroys the d-wave superconducting gap
to uncover a Fermi surface that behaves like a conventional (i.e.Fermi Liquid)
metal (1). Another view is that an applied magnetic field destroys long range
superconducting phase coherence, but the superconducting gap amplitude survives
(2, 3). By measuring the specific heat of ultra-clean YBa2Cu3O6.56 (YBCO 6.56),
we obtain a measure of the quasi-particle density of states from the
superconducting state well into the magnetic-field-induced resistive state. We
have found that at very high magnetic fields the specific heat exhibits both
the conventional temperature dependence and quantum oscillations expected for a
Fermi Liquid. On the other hand, the magnetic field dependence of the
quasi-particle density of states follows a \sqrt{H} behavior that persists
right through the zero-resistance transition, evidencing the fully developed
d-wave superconducting gap over the entire magnetic field range measured. The
coexistence of these two phenomena pose a rigorous thermodynamic constraint on
theories of high-magnetic-field resistive state in the cuprates.
