Exceptional point and hysteresis in perturbations of Kerr black holes

We employ the isomonodromic method to study linear scalar massive perturbations of Kerr black holes for generic scalar masses Mμ and generic black hole spins $a/M$. A degeneracy between the longest-living quasinormal mode and the first overtone is found for (Mμ)c 0.3704981 and (a/M)c 0.9994660. We show that the longest-living mode and the first overtone change continuously into each other as we vary the parameters around the point of degeneracy, providing evidence for the existence of a geometric phase around an exceptional point. We interpret our findings through a thermodynamic analogy.