Multiscale Excitations in the Diluted Two-dimensional S = 1/2 Heisenberg Antiferromagnet

We study the excitation spectrum of the $S=1/2$ Heisenberg model on the randomly diluted square lattice by analytic continuation of QMC data. At dilution fractions $p=1/16$ and $p=1/8$, the dynamic structure factor $S({\bf q},\omega)$ exhibits a damped magnon peak with anomalous dispersion near ${\bf q}=(0,0)$ and $(\pi,\pi)$, a non-dispersive low-energy localization peak, and a second peak between these two features. A magnon with anomalous dispersion, close to our result, was predicted in spin wave and $T$-matrix theory [A. Chernyshev et al., PRB {\bf 65}, 104407 (2002)], above the localization energy. However, no intermediate mode was predicted. Analyzing spectral functions in real space for individual vacancy realizations by energy tomography, we find that these excitations are concentrated on a small subset of the spins adjacent to vacancies. We argue that the low-energy excitations are those of a sparse random network of effective moments at a fraction of the vacancies. There is a shift in magnon spectral weight distribution, from the spins away from vacancies at high energy to those adjacent to vacancies at lower energy. We also analyze the Anderson quantum rotor excitation at $\omega \propto N^{-1}$ (with $N=L^2$ the system size), which in the clean system is visible in $S({\bf q},\omega)$ only at ${\bf q}=(\pi,\pi)$ but spreads through the Brillouin zone when $p>0$. Weight close to ${\bf q}=(0,0)$ and $(\pi,\pi)$ is explained by local sublattice imbalance within a dimer-monomer model but there is also structure arising from correlated singlet fluctuations, which we demonstrate by enhancing said fluctuations with four-spin couplings. All spectral features found here should be observable by elastic neutron scattering experiments on layered quantum antiferromagnets doped with nonmagnetic impurities.