  We use QMC simulations to study effects of disorder on the $S=1/2$ Heisenberg
model with exchange constant $J$ on the square lattice supplemented by
multispin interactions $Q$. It was found recently [L. Lu et al., Phys. Rev. X
8, 041040 (2018)] that the ground state of this $J$-$Q$ model with random
couplings undergoes a quantum phase transition from the N\'eel state into a
randomness-induced spin-liquid-like state that is a close analogue to the well
known random-singlet (RS) state of the random Heisenberg chain. The 2D RS state
arises from spinons localized at topological defects. The interacting spinons
form a critical state with mean spin-spin correlations decaying with distance
$r$ as $r^{-2}$, as in the 1D RS state. The dynamic exponent $z \ge 2$, varying
continuously with the model parameters. We here further investigate the
properties of the RS state in the $J$-$Q$ model with random $Q$ couplings. We
study the temperature dependence of the specific heat and various
susceptibilities for large enough systems to reach the thermodynamic limit and
also analyze the size dependence of the critical magnetic order parameter and
its susceptibility in the ground state. For all these quantities, we find
consistency with conventional quantum-critical scaling when the condition
implied by the $r^{-2}$ form of the spin correlations is imposed. All
quantities can be explained by the same value of the dynamic exponent $z$ at
fixed model parameters. We argue that the RS state identified in the $J$-$Q$
model corresponds to a generic renormalization group fixed point that can be
reached in many quantum magnets with random couplings, and may already have
been observed experimentally.
