Singular points of time-dependent correlation functions of spin systems on large-dimensional lattices at high temperatures

Time-dependent autocorrelation functions are investigated for the Heisenberg model with spins 1/2 on $d$-dimensional simple cubic lattices of large dimensions $d$ at infinite temperature. The autocorrelation function on the imaginary time axis is interpreted as the generating function of bond trees constructed with double bonds. These trees provide the leading terms with respect to $1/d$ for the time-expansion coefficients of the autocorrelation function. The correction terms from branch intersections to the generating function in the Bethe approximation are derived for these trees. A procedure is suggested for finding the correction to the coordinate of the singular point of the generating function (i.e., to the reciprocal of the branch growth-rate parameter) from the above correction terms without calculating the number of trees. The leading correction terms of order $1/\sigma^2$ (where $\sigma=2d-1$) are found for the coordinates of the singular points of the autocorrelation function in question and for the generating function of the trees constructed with single bonds in the Eden model.