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Mott glass phase in a diluted bilayer Heisenberg quantum antiferromagnet

We use quantum Monte Carlo simulations to study a dimer-diluted $S=1/2$ Heisenberg model on a bilayer square lattice with intralayer interaction $J_{1}$ and interlayer interaction $J_{2}$. Below the classical percolation threshold $p_c$, the system has three phases reachable by tuning the interaction ratio $g=J_{2}/J_{1}$: a N$\acute{e}$el ordered phase, a gapless quantum glass phase, and a gapped quantum paramagnetic phase. We present the ground-state phase diagram in the plane of dilution $p$ and interaction ratio $g$. The quantum glass phase is certified to be of the gapless Mott glass type, having a uniform susceptibility vanishing at zero temperature $T$ and following a stretched exponential form at $T>0$; $χ_u \sim \exp(-b/T^α)$ with $α< 1$. At the phase transition point from N$\acute{e}$el ordered to Mott glass, we find that the critical exponents are different from those of the clean system described by the standard $O(3)$ universality class in 2+1 dimensions.