Of Disorder Averaging and Finite Size Scaling

Ferenc Pázmándi, R. T. Scalettar, Gergely T. Zimányi

E. Brézin

Karim Bernardet, Ferenc Pázmándi, G. G. Batrouni

We propose a new picture of the renormalization group (RG) approach in the presence of disorder, which considers the RG trajectories of each random sample (realization) separately instead of the usual renormalization of the averaged free energy. The main consequence of the theory is that the average over randomness has to be taken after finding the critical point of each realization. To demonstrate these concepts, we study the finite-size scaling properties of the two-dimensional random-bond Ising model. We find that most of the previously observed finite-size corrections are due to the sample-to-sample fluctuation of the critical temperature and scaling predictions are fulfilled only by the new average.

Ferenc Pázmándi, Richard T. Scalettar, Gergely T. Zimányi

For phase transitions in disordered systems, an exact theorem provides a bound on the finite size correlation length exponent: ${\ensuremath{\nu}}_{\mathrm{FS}}\ensuremath{\ge}2/d$. It is believed that the intrinsic $\ensuremath{\nu}$ satisfies the same bound. We argue that the standard averaging introduces a noise and a new diverging length scale. For $\ensuremath{\nu}\ensuremath{\le}2/d$ self-averaging breaks down, disconnecting $\ensuremath{\nu}$ from ${\ensuremath{\nu}}_{\mathrm{FS}}$, and the bound applies only for the latter. We illustrate these ideas on two exact examples, with $\ensuremath{\nu}<2/d$. We propose a new method of disorder averaging, which is able to capture the intrinsic exponents.

Vladimir Privman A