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arXiv:cond-mat/9704155 · Physical Review Letters

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

Published Dec 22, 1997• 98 citations
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Abstract

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.

DOI
10.1103/physrevlett.79.5130

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For phase transitions in disordered systems, an exact theorem provides a bound on the finite size correlation length exponent: \ensuremathνFS\ensuremath≥2/d{\ensuremath{\nu}}_{\mathrm{FS}}\ensuremath{\ge}2/d\ensuremathνFS​\ensuremath≥2/d. It is believed that the intrinsic \ensuremathν\ensuremath{\nu}\ensuremathν satisfies the same bound. We argue that the standard averaging introduces a noise and a new diverging length scale. For \ensuremathν\ensuremath≤2/d\ensuremath{\nu}\ensuremath{\le}2/d\ensuremathν\ensuremath≤2/d self-averaging breaks down, disconnecting \ensuremathν\ensuremath{\nu}\ensuremathν from \ensuremathνFS{\ensuremath{\nu}}_{\mathrm{FS}}\ensuremathνFS​, and the bound applies only for the latter. We illustrate these ideas on two exact examples, with \ensuremathν<2/d\ensuremath{\nu}<2/d\ensuremathν<2/d. We propose a new method of disorder averaging, which is able to capture the intrinsic exponents.
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 $\ensuremath{\nu}$, 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.
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.
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.
Abstract To understand the finite-size-scaling properties of phases transitions in classical and quantum models in the presence of quenched disorder, it has proven to be fruitful to introduce the notion of a finite-size-pseudo-critical point in each disordered sample and to analyze its sample-to-sample fluctuations as a function of the size. For the many-body-localization transition, where very strong eigenstate-to-eigenstate fluctuations have been numerically reported even within a given disordered sample at a given energy density (Yu, Luitz and Clark 2016 arXiv: 1606.01260 and Khemani, Lim, Sheng and Huse 2016 arXiv: 1607.05756 ), it seems thus useful to introduce the notion of a finite-size-pseudo-critical point for each individual eigenstate and to study its eigenstate-to-eigenstate fluctuations governed by the correlation length exponent ν . The scaling properties of critical eigenstates are also expected to appear much more clearly if one considers each eigenstate at its finite-size-pseudo-critical point, where it is ‘truly critical’, while standard averages over eigenstates and samples in the critical region actually see a mixture of states that are effectively either localized or delocalized.
We show by numerical simulations that the correlation function of the random-field Ising model (RFIM) in the critical region in three dimensions has very strong fluctuations and that in a finite volume the correlation length is not self-averaging. This is due to the formation of a bound state in the underlying field theory. We argue that this nonperturbative phenomenon is not particular to the RFIM in 3D. It is generic for disordered systems in two dimensions and may also happen in other three-dimensional disordered systems.
We study the T=0 critical properties of the superfluid-insulator transition in 2D hard core Bose systems with disorder. Using quantum Monte Carlo simulations and finite size scaling on 64\ifmmode\times\else\texttimes\fi{}64 lattices we find the dynamical exponent z=0.5\ifmmode\pm\else\textpm\fi{}0.1 and the correlation length exponent \ensuremath{\nu}=2.2\ifmmode\pm\else\textpm\fi{}0.2. At the transition, the system is metallic with a conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathit{c}}$=(1.2\ifmmode\pm\else\textpm\fi{}0.2)(e∗{\mathit{e}}^{\mathrm{*}}e∗)2{)}^{2})2/h and a nonzero compressibility \ensuremath{\kappa}\ensuremath{\ne}0. These conclusions differ from the existing scaling theory as well as from simulations on simplified models argued to be in the same universality class. Our results are suggestive of new low lying collective excitations (modified from usual phonons) in the disordered system.