Unveiling nonmagnetic phase and many-body entanglement in two-dimensional random quantum magnets Sr$_2$CuTe$_{1-x}$W$_x$O$_6$

We apply a random-plaquette $J_1$-$J_2$ model on the square lattice to capture the physics of a series of spin-$1/2$ Heisenberg antiferromagnet compounds Sr$_2$CuTe$_{1-x}$W$_x$O$_6$. With the input of experimentally relevant coupling strengths, our exact diagonalization (ED) study probes the ground state properties beyond previous linear spin-wave approach. An intermediate range of $x \in [0.08, 0.55]$ is identified for a nonmagnetic phase without the long-range N\'eel or stripe order. The absence of both valence-bond-glass order and spin-glass non-ergodic dynamics renders its nature intriguing. Deep inside this phase around $x = 0.3$, we observe signatures potentially linked to randomness-induced short-range spin-liquid-like (SLL) states, including close to zero spin-freezing parameter, vanishing spin-spin correlation beyond nearest neighbors, almost uniform static spin structure factor, as well as a broad tail in the dynamical spin structure factor. The nonmagnetic phase also features multipartite entanglement in the ground state witnessed by quantum Fisher information (QFI), which exhibits universal scaling behaviors at quantum critical points.