Effective Field Theory for Layered Quantum Antiferromagnets with Non-Magnetic Impurities

We propose an effective two-dimensional quantum non-linear sigma model combined with classical percolation theory to study the magnetic properties of site diluted layered quantum antiferromagnets like La${}_2$Cu${}_{1-x}$M${}_x$O${}_4$ (M$=$Zn, Mg). We calculate the staggered magnetization at zero temperature, $M_s(x)$, the magnetic correlation length, $\xi (x,T)$, the NMR relaxation rate, $1\mathrm{/}{T}_1(x,T)$, and the N'eel temperature, $T_N(x)$, in the renormalized classical regime. Due to quantum fluctuations we find a quantum critical point (QCP) at $x_c\approx 0.305$ at lower doping than the two-dimensional percolation threshold $x_p\approx 0.41$. We compare our results with the available experimental data.