The pseudogap in YBa$_2$Cu$_3$O$_{6+\delta}$ is not bounded by a line of phase transitions - thermodynamic evidence
We discuss a recent resonant ultrasound spectroscopy (RUS) study of YBa$_2$Cu$_3$O$_{6+\delta}$, which infers a line of phase transitions bounding the pseudogap phase and argue that this scenario is not supported by thermodynamic evidence. We show that the anomalies in RUS, heat capacity and thermal expansion at the superconducting transition temperatures agree well. But there are large discrepancies between RUS and thermodynamic measurements at $T^*$ where the pseudogap phase transitions are purported to occur. Moreover, the frequency and temperature dependence of the RUS data for the crystal with $\delta = 0.98$, interpreted in terms of critical slowing down near an electronic phase transition, is five orders of magnitude smaller than what is expected. For this crystal the RUS data near $T^*$ are more consistent with non-equilibrium effects such as oxygen relaxation.