# Does the critical dynamical exponent z of a 2D Ising model (simulated with Metropolis) vary with the temperature?

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?

• 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. – Christophe Feb 22 '19 at 9:21
• @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? – lucia de finetti Feb 22 '19 at 10:31

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.