Abstract
A first-order phase transition is found in the multilayer cuprate superconductor, HgBa$_2$Ca$_4$Cu$_5$O$_y$ (Hg-1245), with a superconducting transition temperature of 108 K, under zero magnetic field. We observed a hysteretic specific heat jump around 41 K. We conclude that the Bardeen-Cooper-Schrieffer pairs have a residual entropy due to fluctuations in the phase difference between the five CuO$_2$ planes in a unit cell of Hg-1245, and that this fluctuation freezes below the first-order phase transition temperature.
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