Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Finite-Size Scaling and Correlation Lengths for Disordered Systems

Jennifer Chayes, L. Chayes, Daniel S. Fisher, T. Spencer

Published Dec 15, 1986• 674 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.57.2999

Related papers

Revisiting the Theory of Finite Size Scaling in Disordered Systems: ν Can Be Less than 2/d

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.

arXiv:cond-mat/9704155•Physical Review Letters•Dec 22, 1997•98 citations

Finite-size correlation length and violations of finite-size \nscaling

Sergio Caracciolo A, Andrea Gambassi, Massimiliano Gubinelli +1 more

We address the problem of the definition of the finite-volume correlation length. First, we study the large-N limit of the N-vector model, and we show the existence of several constraints on the definition if regularity of the finite-size scaling functions and correct anomalous behaviour above the upper critical dimension are required. Then, we study in detail a model in which the zero mode is prohibited. Such. a model is a generalization of the fixed-magnetization Ising model which is equivalent to the lattice gas. Also in this case, we find that the finite-volume correlation length must satisfy appropriate constraints in order to obtain regular finite-size scaling functions, and, above the upper critical dimension, an anomalous scaling behaviour. The large-N results are confirmed by a one-loop calculation in the lattice phi (4) theory.

•Jan 1, 2001•31 citations

Critical exponent and multifractality of states at the Anderson transition

Bernhard Krämer

Based on the assumptions of the existence of a one-paramter scaling law for a self-averaging scaling variable, and on a finite correlation length for the randomness, the lower bound for the critical exponent of the Anderson transition derived earlier by Chayes et al. [Phys. Rev. Lett. 57, 2999 (1986)] for uncorrelated disorder is generalized to the case of spatially correlated disorder. The bound is determined by the system-size dependence of the number of the random variables that are necessary to describe the transition. It is speculated how to relate the critical exponent to the multifractal properties of the states near the critical point. Exponents of several models are estimated and compared with numerical scaling calculations.

Physical review. B, Condensed matter•Apr 15, 1993•15 citations

Finite-size effects in one-dimensional percolation: a verification of scaling theory

Stanley Greenspoon

The critical behaviour of the one-dimensional percolation problem is studied for a large, but finite, number of sites, N, in the light of finite-size scaling theory. The functions of interest, G(p), P(p), and S(p), are shown to be expressible asymptotically in a form consistent with scaling theory, in terms of a scaling parameter proportional to N/ξ, ξ being the correlation length. A plausible quasi-critical percolation probability for finite systems is proposed, and quasi-criticality is shown to occur at a unique value of the scaling parameter, in accord with a previously discussed law of corresponding states for finite systems. The rounding and shift exponents are found to be given by θ = λ = 1.

Canadian Journal of Physics•Apr 1, 1979•3 citations

Finite-Size Scaling and Critical Exponents. A New Approach and Its Application to Anderson Localisation

Etienne Hofstetter, Michael Schreiber

Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.

Europhysics Letters (EPL)•Mar 20, 1993•41 citations

Universality in Anderson localization on random graphs with varying connectivity

Piotr Sierant, Maciej Lewenstein, Antonello Scardicchio

We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder $W_{E}$ which is strictly less than the Anderson localization critical disorder $W_C$, and that of other works which suggest that there is no such region. We find that, although no separate $W_{E}$ exists from $W_C$, the length scale at which fully developed ergodicity is found diverges like $|W-W_C|^{-1}$, while the critical length over which delocalization develops is $\sim |W-W_C|^{-1/2}$. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.

arXiv:2205.14614•May 29, 2022•1 citation
For a large class of ddd-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 \ensuremathν\ensuremath{\nu}\ensuremathν, then \ensuremathν\ensuremath{\nu}\ensuremathν must satisfy the bound \ensuremathν\ensuremath≥2d\ensuremath{\nu}\ensuremath{\ge}\frac{2}{d}\ensuremathν\ensuremath≥d2​. 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 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.
We address the problem of the definition of the finite-volume correlation length. First, we study the large-N limit of the N-vector model, and we show the existence of several constraints on the definition if regularity of the finite-size scaling functions and correct anomalous behaviour above the upper critical dimension are required. Then, we study in detail a model in which the zero mode is prohibited. Such. a model is a generalization of the fixed-magnetization Ising model which is equivalent to the lattice gas. Also in this case, we find that the finite-volume correlation length must satisfy appropriate constraints in order to obtain regular finite-size scaling functions, and, above the upper critical dimension, an anomalous scaling behaviour. The large-N results are confirmed by a one-loop calculation in the lattice phi (4) theory.
Based on the assumptions of the existence of a one-paramter scaling law for a self-averaging scaling variable, and on a finite correlation length for the randomness, the lower bound for the critical exponent of the Anderson transition derived earlier by Chayes et al. [Phys. Rev. Lett. 57, 2999 (1986)] for uncorrelated disorder is generalized to the case of spatially correlated disorder. The bound is determined by the system-size dependence of the number of the random variables that are necessary to describe the transition. It is speculated how to relate the critical exponent to the multifractal properties of the states near the critical point. Exponents of several models are estimated and compared with numerical scaling calculations.
The critical behaviour of the one-dimensional percolation problem is studied for a large, but finite, number of sites, N, in the light of finite-size scaling theory. The functions of interest, G(p), P(p), and S(p), are shown to be expressible asymptotically in a form consistent with scaling theory, in terms of a scaling parameter proportional to N/ξ, ξ being the correlation length. A plausible quasi-critical percolation probability for finite systems is proposed, and quasi-criticality is shown to occur at a unique value of the scaling parameter, in accord with a previously discussed law of corresponding states for finite systems. The rounding and shift exponents are found to be given by θ = λ = 1.
Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.
We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder WEW_{E}WE​ which is strictly less than the Anderson localization critical disorder WCW_CWC​, and that of other works which suggest that there is no such region. We find that, although no separate WEW_{E}WE​ exists from WCW_CWC​, the length scale at which fully developed ergodicity is found diverges like ∣W−WC∣−1|W-W_C|^{-1}∣W−WC​∣−1, while the critical length over which delocalization develops is ∼∣W−WC∣−1/2\sim |W-W_C|^{-1/2}∼∣W−WC​∣−1/2. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.