I have found in the literature that the critical dynamical exponent $z$ of an Ising model simulated with a local algorithm (such as Metropolis) is something around 2 near the critical temperature, where the critical slowing-down occurs and, luckily, that is also true for my personal simulations near the critical temperature 2.269..?

Still, when I go at temperature higher than the critical one, I find an higher exponent z. Is that normal? And if so, why's that?

  • $\begingroup$ It seems to me that there is no reason anymore for the autocorrelation time to increase algebraically with the lattice size as $L^z$ outside the critical point. I would expect that it saturates at some limiting value in the paramagnetic phase. $\endgroup$
    – Christophe
    Feb 22, 2019 at 9:21
  • $\begingroup$ @Christophe hi and thanks for the reply! The autocorrelation time should actually decrease while going away from the critical region to higher temperatures so yeah, I agree! And what about z in your opinion? $\endgroup$ Feb 22, 2019 at 10:31

1 Answer 1


Like any other critical exponent, the dynamical exponent $z$ can be extracted from the scaling of the autocorrelation time $\tau$ with

  • the reduced temperature as $\tau\sim |T-T_c|^{-z}$ for $T$ sufficiently close to the critical temperature
  • the lattice size $L$ as $\tau\sim L^z$ only at $T=T_c$.

Away from the critical point, the time $\tau$ does not diverge anymore. Therefore, there is no point in fitting $\tau$ as $\tau\sim L^z$ for $T\ne T_c$ in the Ising model. Only in a critical phase (as in the 2D-XY model), or in glasses, there exists a temperature-dependent dynamical exponent.

  • $\begingroup$ What I tought, thank you very much! Any book that discusses this? $\endgroup$ Feb 23, 2019 at 1:27

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