Finite-temperature phase transitions in $S=1/2$ three-dimensional Heisenberg magnets from high-temperature series expansions

By means of high-temperature series expansions we study the singularities of the specific heat ($c_v$) and the magnetic susceptibility ($\chi$) of $S=1/2$ models presenting a phase transition belonging to the three dimensional Heisenberg universality class. We first calculate the critical temperature $T_c$ and the critical exponent $\gamma$ using the standard Dlog Pad\'e method on $\overline{\chi}(\beta)=\chi(\beta)/\beta$ for the ferromagnetic Heisenberg model on the face-centered cubic, body-centered cubic, simple cubic, pyrochlore and semi-simple cubic lattices ($\beta=1/T$). We also explore the possibility of using this method on $\overline{\chi}(e)$ and $c_v(\beta)$ to calculate the critical energy $e_c$ and the critical exponent $\alpha$. Finally, we adapt a method initially developed for logarithmic singularities [Phys. Rev. B 104, 165113 (2021)] to cusp ($-1<\alpha<0$) and divergent singularities ($\gamma>0$), and propose different interpolation methods for $c_v$ and $\chi$. We apply our method to several of the previously mentioned lattices and present for each the reconstructed $c_v$ and $\chi$ down to $T_c$.