Critical nonequilibrium relaxation in cluster algorithms using the Binder ratio and its application to bond-diluted Ising models

Recently we showed that the critical nonequilibrium relaxation in cluster algorithms is widely described by the stretched-exponential relaxation [1-3]. Explicitly, the absolute value of magnetization at the critical temperature $T_{\rm c}$ behaves as $\langle |m| \rangle \sim \exp (+c_{m}t^{\sigma}$) from the perfectly-disordered state. In the present talk we apply this scheme to the bond-diluted Ising models and show that the exponent $\sigma$ increases continuously and monotonously as the bond density $p$ decreases. Although na\"ive fitting of physical quantities becomes difficult as $p$ approaches the percolation threshold $p_{\rm c}$, we find that the Binder ratio has no such a problem even in the vicinity of $p_{\rm c}$. While the Binder ratio is almost independent of system sizes at $T_{\rm c}$ both at the onset of relaxation and near equilibrium, the exponent $\sigma$ can be estimated accurately by an empirical logarithmic scaling for the size dependence in the intermediate simulation-time region. \smallskip \par \noindent [1] Y.~Nonomura, J.\ Phys.\ Soc.\ Jpn.\ {\bf 83}, 113001 (2014). [2] Y.~Nonomura and Y.~Tomita, Phys.\ Rev.\ E {\bf 92}, 062121 (2015). [3] Y.~Nonomura and Y.~Tomita, Phys.\ Rev.\ E {\bf 93}, 012101 (2016).