Specific heat and sound velocity at the relevant competing phase of high-temperature superconductors

Recent highly accurate sound velocity measurements reveal a phase transition to a competing phase in YBa2Cu3O6+δ that is not identified in available specific heat measurements. We show that this signature is consistent with the universality class of the loop current-ordered state when the free-energy reduction is similar to the superconducting condensation energy, due to the anomalous fluctuation region of such a transition. We also compare the measured specific heat with some usual types of transitions, which are observed at lower temperatures in some cuprates, and find that the upper limit of the energy reduction due to them is about 1/40th the superconducting condensation energy.