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.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.

arXiv:cond-mat/9704155Physical Review LettersDec 22, 199798 citations

Recent advances in the theory of disordered systems

Ferenc Iglói, Heiko Rieger

The European Physical Journal BMay 1, 20206 citations

Finite-Size Scaling and Correlation Lengths for Disordered Systems

Jennifer Chayes, L. Chayes, Daniel S. Fisher +1 more

For a large class of $d$ -dimensional disordered systems, we prove that if an appropriately defined finite-size scaling correlation length diverges at a nontrivial value of the disorder with an exponent $\ensuremath{\nu}$, then $\ensuremath{\nu}$ must satisfy the bound $\ensuremath{\nu}\ensuremath{\ge}\frac{2}{d}$. Given the assumption that such a correlation length can be defined, the result applies to, e.g., percolation, disordered magnets, and Anderson localization, both with and without interactions. For localization, this puts stringent constraints on scaling theories and interpretation of experiments.For a large class of $d$-dimensional disordered systems, we prove that if an appropriately defined finite-size scaling correlation length diverges at a nontrivial value of the disorder with an exponent $\ensuremath{\nu}$, then $\ensuremath{\nu}$ must satisfy the bound $\ensuremath{\nu}\ensuremath{\ge}\frac{2}{d}$. Given the assumption that such a correlation length can be defined, the result applies to, e.g., percolation, disordered magnets, and Anderson localization, both with and without interactions. For localization, this puts stringent constraints on scaling theories and interpretation of experiments.

Physical Review LettersDec 15, 1986674 citations

Density of states and finite size scaling investigations

Ramon Villanova, Nelson A. Alves, Bernd A. Berg

Nuclear Physics B - Proceedings SupplementsMay 1, 19914 citations

Many-body delocalization in a strongly disordered system with long-range interactions: Finite-size scaling

Alexander L. Burin

Many-body localization in a disordered system of interacting spins coupled by the long-range interaction $1/{R}^{\ensuremath{\alpha}}$ is investigated combining analytical theory considering resonant interactions and a finite-size scaling of exact numerical solutions with number of spins $N$ . The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for a $d$ -dimensional system including the absence of localization in the infinite system at $\ensuremath{\alpha}<2d$ and a universal scaling of a critical energy disordering ${W}_{c}\ensuremath{\propto}{N}^{\frac{2d\ensuremath{-}\ensuremath{\alpha}}{d}}$.Many-body localization in a disordered system of interacting spins coupled by the long-range interaction $1/{R}^{\ensuremath{\alpha}}$ is investigated combining analytical theory considering resonant interactions and a finite-size scaling of exact numerical solutions with number of spins $N$. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for a $d$-dimensional system including the absence of localization in the infinite system at $\ensuremath{\alpha}<2d$ and a universal scaling of a critical energy disordering ${W}_{c}\ensuremath{\propto}{N}^{\frac{2d\ensuremath{-}\ensuremath{\alpha}}{d}}$.

arXiv:1410.7858Physical Review BMar 4, 2015122 citations

For a large class of

d

-dimensional disordered systems, we prove that if an appropriately defined finite-size scaling correlation length diverges at a nontrivial value of the disorder with an exponent $\ensuremath{\nu}$, then $\ensuremath{\nu}$ must satisfy the bound $\ensuremath{\nu}\ensuremath{\ge}\frac{2}{d}$. Given the assumption that such a correlation length can be defined, the result applies to, e.g., percolation, disordered magnets, and Anderson localization, both with and without interactions. For localization, this puts stringent constraints on scaling theories and interpretation of experiments.

N

. The numerical results for a one-dimensional system are consistent with the general expectations of analytical theory for a

d

-dimensional system including the absence of localization in the infinite system at $\ensuremath{\alpha}<2d$ and a universal scaling of a critical energy disordering ${W}_{c}\ensuremath{\propto}{N}^{\frac{2d\ensuremath{-}\ensuremath{\alpha}}{d}}$.