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.
- Tom Bienaimé and Christophe Texier,
Localization for one-dimensional random potentials with large local fluctuations
J. Phys. A: Math. Theor. 41, 475001 (2008). (9pp)
cond-mat arXiv:0807.0772.
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:
- Kabir Ramola and Christophe Texier,
Fluctuations of random matrix products and 1D Dirac equation with random mass,
J. Stat. Phys. 157, 497-514 (2014)
cond-mat arXiv:1402.6943
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