Dispersion of a single hole in an antiferromagnet

Abstract

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 “noncrossing” Born approximation. We found that for $t/J\gg 1$, the bare fermionic dispersion is completely overshadowed by the self-energy corrections. In this case, the quasiparticle Green’s function contains a broad incoherent continuum which extends over a frequency range of $\sim 6t$. In addition, there exists a narrow region of width $O\left(\mathrm{JS}\right)\mathrm{}$ below the top of the valence band, where the excitations are mostly coherent, though with a small quasiparticle residue $Z\sim J/t$. The top of the valence band is located at $(\pi /2,\pi /2)$. We found that the form of the fermionic dispersion, and, in particular, the ratio of the effective masses near $(\pi /2,\pi /2)$ strongly depend on the assumptions one makes for the form of the magnon propagator. 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 ${\mathrm{}\left(4JS\right)\mathrm{}}^{-1}$. We further computed the full fermionic dispersion for $J/t=0.4\mathrm{}$ relevant for ${\mathrm{Sr}}_2{\mathrm{CuO}}_2{\mathrm{Cl}}_2$, and $t^{\prime }=-0.4J$ and found not only that the masses are both equal to ${\mathrm{}\left(2J\right)\mathrm{}}^{-1}$, but also that the energies at $\mathrm{}(0,0)\mathrm{}$ and $(0,\pi )$ are equal, the energy at $(0,\pi /2)$ is about half of that at $\mathrm{}(0,0)\mathrm{}$, and the bandwidth for the coherent excitations is around $3J$. All of these results are in full agreement with the experimental data. Finally, we found that weakly damped excitations only exist in a narrow range around $(\pi /2,\pi /2)$. Away from the vicinity of $(\pi /2,\pi /2)$, the excitations are overdamped, and the spectral function possesses a broad maximum rather than a sharp quasiparticle peak. This last feature was also reported in photoemission experiments.