Fluctuations and single parameter scaling

Localization properties with potential with large local fluctuations

Most of works on 1D Anderson localisation consider the case where the potential has relatively small local fluctuations, such that <Vn 2> < ∞ (Vn is the on-site potenial for discrete models) or ∫-∞+∞dx <V(x)V(0)> < ∞ (for continuous models). Models where this condition is not fulfilled lead to non standard localization properties (super-localisation) with non exponential damping of wave function’s envelope, like exp(-|x|α) for α>1.


Fluctuations of random matrix products of SL(2,R) and localisation in the random mass Dirac equation

We study the fluctuations of the logarithm of the wave functions, a problem related to the analysis of the fluctuations of certain random matrix products:

This question is of importance in localisation problems : it is related to the discussion of the Single Parameter Scaling hypothesis ; this plays an important role when studying statistical properties of local density of states (Altshuler & Prigodin, 1989) or Wigner time delay (Texier & Comtet, Phys. Rev. Lett. 82(21), 4220 (1999)).

See also page “ products of random matrices

Fluctuations and single parameter scaling for 1D disorder

The « single parameter scaling » (SPS) hypothesis is a corner stone of the scaling theory of localization. It states that the full distribution of observable is controlled by a single characteristic scale (the localization length). First discussed within models with ad hoc random phase assumption (see the nice article: Cohen, Roth & Shapiro, Phys. Rev. B 38, 12125 (1988)), the solvable Lloyd model has provided a ground to test SPS within a microscopic model (Deych, Lisyansky & Altshuler, Phys. Rev. Lett. 84, 2678 (2000)).
In the following paper, a general formula for the variance of ln|ψ(x)| is obtained, where ψ(x) solves the Schrödinger equation, for arbitrary disorder characterised by its Lévy exponent L(s):

  • Christophe Texier,
    Fluctuations of the product of random matrices and generalized Lyapunov exponent,
    J. Stat. Phys (2020)
    cond-mat arXiv:1907.08512
    Some integral formula is derived for γ2=limx→∞(1/x)Var(ln|ψ(x)|) for the Schrödinger equation with a random potential.

See also page “ products of random matrices

Using this general formalism, I have provided in the previous paper a general framework allowing to analyse SPS in a very broad class of models with both finite of infinite second moment (like for the Lloyd model). A universal formula for the generalised Lyapunov exponent (cumulant generating function of ln|ψ(x)|) is derived:

  • Christophe Texier,
    Generalized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation,
    Europhys. Lett. 131, 17002 (2020)
    cond-mat arXiv:1910.01989
    Using the general formalism of the previous article, the Single Parameter Scaling for Anderson localization is proven within a general framework, and extended to potentials with large fluctuations (such that <V2>=∞)
  • Alain Comtet, Christophe Texier & Yves Tourigny,
    The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results,
    cond-mat arXiv:2110.01522
    Taking advantage of the specificity of the Lloyd model, we are able to get a secular equation for the generalized Lyapunov exponent, from which we derive a set of exact results

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