Single-particle excitations in a quantum antiferromagnet

Results for the single-particle Green's function for the strong-coupling limit of the single-band Hubbard model ($t-J$ model) obtained by numerical and analytical methods are discussed. The numerical calculation of the spectral function is performed for small systems of up to 18 sites using the Lanczos method. A detailed analysis of the spectral functions for the $t-J$ model in one and two dimensions is given. We also investigate the effect of the next-nearest-neighbor hopping terms, which appear in order $\frac{t^2}{U}$ and are omitted in the standard $t-J$ model. The role of quantum fluctuations on the shape of the spectra and the fundamental difference between one and two dimensions is elucidated. In 2D we find evidence for a coherent quasiparticle band of width $2J$ with dispersion ${}_{}{}^{\omega }{}_{}{}^{}\mathrm{k}\sim \left(\frac{J}{2}\right){(cos{k}_x+cos{k}_y)}^2$ and spin $S=\frac{1}{2}$, which emerges at the low-energy edge of the incoherent spectrum. This form of dispersion is due to effective next-nearest-neighbor processes related to spin flips and agrees with our extension of the approach of Brinkman and Rice to finite $J$.