The dispersion of a single hole in an antiferromagnet

We revisit the problem of the dispersion of a single hole injected into a quantum antiferromagnet. We applied a spin-density-wave formalism extended to a large number of orbitals and obtained an integral equation for the full quasiparticle Green's function in the self-consistent "non-crossing" Born approximation. We found that for $t/J \gg 1$, the bare fermionic dispersion is completely overshadowed by the self-energy corrections. We obtain a broad incoherent continuum which extends over a frequency range of $\sim 6t$ and a narrow region of width $O(JS)$ below the top of the valence band, where the excitations are mostly coherent, though with a small quasiparticle residue $Z \sim J/t$. Furthermore, we argue in this paper that two-magnon Raman scattering as well as neutron scattering experiments strongly suggest that the zone boundary magnons are not free particles since a substantial portion of their spectral weight is transferred into an incoherent background. We modeled this effect by introducing a cutoff $q_c$ in the integration over magnon momenta. We found analytically that for small $q_c$, the strong coupling solution for the Green's function is universal, and both effective masses are equal to $(4JS)^{-1}$. We further computed the full fermionic dispersion for $J/t =0.4$ relevant for $Sr_2CuO_2Cl_2$, and $t^\prime=-0.4J$ and found not only that the masses are both equal to $(2J)^{-1}$, but also that the energies at $(0,0)$ and $(0,\pi)$ are equal, the energy at $(0,\pi/2)$ is about half of that at $(0,0)$, and the bandwidth for the coherent excitations is around $3J$. All of these results are in full agreement with the experimental data.