Effect of next-nearest-neighbor hopping on the spin dynamics in antiferromagnets

Recently, inelastic-neutron-scattering (INS) experiments on the insulating parent compounds of high-$T_c$ materials were analyzed to extract the value of the superexchange constant $J.\mathrm{}$ The starting point of the analysis was the nearest-neighbor Heisenberg model. Motivated by recent angle-resolved photoemission experiments, we consider the effects of a next-nearest-neighbor hopping, $t^{\prime },$ in the strong-coupling limit of the spin-density-wave formalism, where it leads to an antiferromagnetic exchange $J^{\prime }>0\mathrm{}$ between next-nearest-neighbor spins. We show that a finite $J^{\prime }$ (a) changes the spin-wave velocity and thus the value of $J$ extracted from the INS data, yielding almost identical values for $J$ in ${\mathrm{La}}_2{\mathrm{CuO}}_4$ and ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.15},$ and (b) leads to a considerable change in the high-frequency spin-dynamics. Finally, we calculate the $1/S$ quantum corrections in the presence of $J^{\prime }$ and discuss their relevance for the INS data.