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Since the mathematical description of boundary-separating gas and liquid states by van der Waals, the critical endpoint (CEP) has garnered the attention of many researchers because of its intriguing nature. At high temperatures and pressures, water achieves a CEP, in which the gas and liquid phases coexist harmoniously. An anomaly in the compressibility emerges in the water CEP. In magnetic materials, CEP appears as a function of temperature and the magnetic field. The magnetic susceptibility shows an anomaly in the magnetic CEP. Gas-liquid and magnetic CEPs are assumed to be similar. Nevertheless, the presence of non-magnetic fluctuations is unanticipated in magnetic CEPs.

UCoAl, which is an itinerant paramagnet, is located near both the tri-critical point and the quantum critical point (QCP). At ambient pressure, a CEP was observed at approximately 10 K and 0.9 T. In this study, ultrasonic sound velocities ranging from 12 to 106 MHz were measured, and the elastic constant C₃₃, which represents the strain susceptibility for elastic strain (displacement along the hexagonal axis), was obtained. We observed a singularity-like elastic softening toward the CEP and discovered that this anomaly cannot be attributed to magnetostrictive coupling. Power law analyses indicate that it originates from electric quadrupoles.

Furthermore, C₃₃exhibited a fascinating ultrasonic dispersion effect, in which the elastic anomaly diminished as the frequency increased. The recorded relaxation time of 3.5 × 10^-8 s at the CEP is the lowest value reported thus far for solids. This value approached the maximum value, at which the crystal collapsed, thus indicating significant structural fluctuations.

Based on NMR studies, UCoAl exhibits pronounced longitudinal spin fluctuations related to the C₃₃ anomaly. The crystal symmetry of UCoAl remains unchanged at CEPs, and maintains the symmetry of the crystal. Fluctuations in the magnetic and structural properties that retain symmetry coevolve toward the CEP. The unconventional slow dynamics indicate the existence of an exotic ground state in this system, which is believed to emerge near the CEP in distinct and extreme environments.

Recently, peculiar superconducting and quadrupolar states were observed in URhGe, UTe₂, and URhSn. The distinct features of UCoAl near the CEP are speculated to be associated with the enigmatic properties of these compounds.

CEPs contribute to various fields ranging from industrial processes to particle physics. Despite extensive research, the nature of CEPs remains unclear. CEPs can potentially reveal novel and unusual physics, thus presenting an interesting avenue for investigating undisclosed facets of matter.

Written by M. Yoshizawa on behalf of all the authors.