On convergence radius of time-power series for spin correlation functions of the Heisenberg magnet at high temperature

The convergence of series in powers of time for spin autocorrelation functions of the Heisenberg magnet are investigated at infinite temperatures on lattices of different dimensions $d$. The calculation data available at the present time for the coefficients of these series are used to estimate the corresponding radii of convergence, whose growth with decreasing $d$ is revealed and explained in a self-consistent approximation. To this end, a simplified nonlinear equation corresponding to this approximation is suggested and solved for the autocorrelation function of a system with an arbitrary number $Z$ of nearest neighbors. The coefficients of the expansion in powers of time for the solution are represented in the form of trees on the Bethe lattice with the coordination number $Z$. A computer simulation method is applied to calculate the expansion coefficients for trees embedded in square, triangular, and simple cubic lattices under the condition that the intersection of tree branches is forbidden. It is found that the excluded volume effect that manifests itself in a decrease in these coefficients and in an increase in the coordinate and exponent of the singularity of the autocorrelation function on the imaginary time axis is intensified with decreasing lattice dimensions.