The randomly diluted quantum Heisenberg antiferromagnet at zero temperature

We use linear spin wave theory and series expansion method to study the diluted quantum antiferromagnetic Heisenberg model at zero temperature near the p = 1 and p = $p\sb{Q}$ respectively, where p is the concentration of the spins or bonds, and $p\sb{Q}$ is the transition point from order to disorder of the quantum spin system. For the first case, our complete calculation to the first order in x $\equiv$ 1 $-$ p gives a dispersion relation for long-wavelength spin waves in the form $\omega(\vec q)$ = $c(x)q + i\gamma(x)q\sp\tau$. Our results for $c(x)$ agrees with previous work, where $c(x)$ = $c(0)\lbrack 1 - \alpha x + O(x\sp2)\rbrack$. In particular the coefficient $\alpha$, which can be related to the helicity modulus and the uniform perpendicular susceptibility, diverges in the limit $d\to 2$. One major new results is that $\tau$ = $d - 1$, which means that d = 2 is the lower critical dimension at which infrared divergences affect the dynamical response due to vacancies. For the second case, the critical behavior of the quantum system near $p\sb{Q}$ is found to be different from that of the percolation and pure systems. Thus there exists a new fixed point in the phase diagram, which specifies a new universality class. The correlation length exponents $\nu$ (which comes from the relation $\xi\sim(p\sb{Q} - p)\sp\nu$, where $\xi$ is the correlation length) are estimated, which are 0.65 $\pm$ 0.2 and 0.55 $\pm$ 0.2 in 2 and 3 dimensions respectively. The exponents of crossover with temperature and time are determined. We also find evidence that $p\sb{Q} \u3e p\sb{c}$