Investigation of soft-mode gapping about a quantum critical point in a disordered Ising magnet

Disorder is known to have profound effects on strongly disordered systems, modifying the critical exponents of phase transitions and potentially suppressing or broadening the transition itself. In this study, we investigate the ferromagnetic-paramagnetic quantum phase transition in the disordered Ising magnet LiHoₓY₁₋ₓF₄ (x = 0.65) using microwave spectroscopy. A magnetic field is applied transverse to the Ising axis to drive the system toward the quantum critical point, while a frequency-tunable microwave loop-gap resonator probes the electronuclear modes near this transition. We focus on the behavior of the electronuclear modes, with particular emphasis on the lowest energy level, which softens near the quantum critical point in pure LiHoF₄ (x = 1). In contrast to pure LiHoF₄, where the fields of Ho³⁺ ions cancel due to crystal symmetry, the substitution of Y³⁺ disrupts this symmetry, inducing random fields at different sites in LiHo_0.65Y_0.35F₄. Our results probe how site-specific random fields experienced by the Ho³⁺ ions can gap the soft mode at the approach to the quantum phase transition.