Probing critical spin fluctuations with a composite magnetoelectric method: A case study on a Kitaev spin liquid candidate Na$_3$Co$_2$SbO$_6$

In correlated quantum materials, divergent critical fluctuations near the quantum critical point are often closely associated with exotic quantum phases of matter, such as unconventional superconductivity and quantum spin liquids. Here we present a simple yet highly sensitive composite magnetoelectric (ME) method for detecting the critical spin fluctuations in quantum magnets. The ME signal is proportional the magnetostriction coefficient, which directly probes the product of magnetization and spin-spin correlation. As a demonstration, the composite ME method is applied to a Kitaev quantum spin liquid candidate Na$_3$Co$_2$SbO$_6$, which shows signs of magnetic field-induced quantum criticality. Notably, the ME signal prominently diverges at the magnetic field-induced tricritical points, particularly at a tricritical point that lies in close proximity to a zero-temperature quantum critical point. A crucial aspect of these tricritical points is their tunability through the modification of the in-plane magnetic field's direction. The direction of magnetic field can thus serve as a handful yet important tuning parameter, alongside pressure and chemical doping, for searching quantum critical points in quantum magnets with pronounced magnetic anisotropy.