Frequency dependence of temporal spin stiffness and short-range magnetic order in the doped two-dimensional Hubbard model

We study the doping and temperature dependencies of temporal and spatial spin stiffnesses of the Hubbard model within the mean-field approach for incommensurate magnetic order. We show that the frequency dependence of temporal spin stiffness within the considered mean-field approach is crucial to obtain small values of correlation length, comparable to those observed in cuprates. Using the obtained spin stiffnesses, we obtain the temperature and doping dependencies of the correlation length within the large-$N$ limit of the respective nonlinear sigma model. In agreement with the experimental data on ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ we obtain short-range magnetic order with a relatively small correlation length at $0.1\ensuremath{\lesssim}x\ensuremath{\lesssim}0.2$, and a magnetically ordered ground state in the narrow doping region $0.05\ensuremath{\lesssim}x\ensuremath{\lesssim}0.1$. The latter state may correspond to the spin-frozen state, observed in the experimental data on ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$.