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We study the spin excitation spectra of the two-dimensional spin-$1/2$Heisenberg model with a checkerboard structures using stochastic analyticcontinuation of the imaginary-time correlation function obtained from a quantumMonte Carlo simulation. The checkerboard models have two differentantiferromagnetic nearest-neighbor interactions $J_{1}$ and $J_{2}$, and thetuning parameter $g$ is defined as $J_{2}/J_{1}$. The dynamic spin structurefactors are systematically calculated in all phases of the models as well as atthe critical points. To give a full understanding of the dynamic spectra, spinwave theory is employed to explain some features of numerical results,especially for the low-energy part. When $g$ is close to $1$, the features ofthe spin excitation spectra of each checkerboard model are roughly the same asthose of the original square lattice antiferromagnetic Heisenberg model, andthe high-energy continuum among them is discussed. In contrast to the othercheckerboard structures investigated in this paper, the $3\times 3$checkerboard model has distinctive excitation features, such as a gap between alow-energy gapless branch and a gapped high-energy part that exists when $g$ issmall. The gapless branch in this case can be regarded as a spin wave inN$\mathrm{\acute{e}}$el order formed by a "block spin" in each $3\times 3$plaquette with an effective exchange interaction originating fromrenormalization. One unexpected finding is that the continuum also appears inthis low-energy branch.

Spin excitation spectra of the two dimensional $S=1/2$ Heisenberg model with a checkerboard structure