Quantum-critical scaling at the Bose-glass transition of the $3d$ diluted Heisenberg antiferromagnet in a field

The nature of the superfluid-to-Bose-glass (SF-BG) quantum phase transition, occurring in systems of interacting bosons immersed in a disordered environment, remains elusive. One fundamental open question is whether or not the transition obeys conventional scaling at quantum critical points (QCPs): this scaling would lock the value of the crossover exponent $\varphi$—dictating the vanishing of the superfluid critical temperature upon approaching the QCP—to the value of quantum critical exponents for the ground-state transition. Yet such a relation between exponents has been called into question by several numerical as well as experimental results on the SF-BG transition. Here we revisit this issue in the case of the $S=1/2$ Heisenberg antiferromagnet on a site-diluted cubic lattice, which lends itself to efficient quantum Monte Carlo simulations. Our results show that the model exhibits a percolation transition in zero applied field, with the correlation length exponent $\nu =0.87\left(8\right)$ and $\varphi =1.1\left(1\right)$ consistent with $3d$ percolation. When applying a sufficiently strong magnetic field, the dilution-induced transition decouples from geometric percolation, and it becomes a SF-BG transition; nonetheless, the $\nu$ and $\varphi$ exponents maintain values consistent with those of the percolation transition. These results contradict the conventional scaling, which predicts $\varphi ⩾2$; and they suggest a possible relationship between the SF-BG transition and percolation of phase coherence.