Localization length in Anderson localized systems

Localization in one dimension is often studied with techniques from random matrix theory. In particular, one can find the Lyapunov exponent corresponding to the random matrices that describe a given model, and under certain circumstances (described at great length in the first link below) this exponent is the inverse of the localization length.

With that in mind, here are estimates of the localization length for two models prevalent in the literature, which use this Lyapunov exponent technique.

First, the Anderson model. This is a tight-binding model of the form (in 1D): $$H=sum_{n} varepsilon_{n}|nranglelangle n|+V sum_{n}| nranglelangle n+1|+h.c. $$ with random disorder $epsilon_n$.

For a disorder drawn randomly between $-W/2$ and $W/2$ (a uniform distribution), and in the middle of the band, the localization length $xi$ in units of the lattice constant is estimated (in the second reference below) as:

$$xi^{-1}(E) = gamma(E)=frac{leftlanglevarepsilon^{2}rightrangle}{V^{2}} Fleft(frac{E V}{leftlanglevarepsilon^{2}rightrangle}right) quad(W,|E| ll V)$$ where $F$ is a smooth function that goes from about 8.7 at $F(0)$ to 8 at $F(infty)$. Actually, I think this is valid for other disorder distributions characterized by some width $W$, but it isn't completely clear to me what the necessary conditions are for this distribution. There is also a more complex scaling relation near the edges of the band:

$$gamma(E)=frac{leftlanglevarepsilon^{2}rightrangle^{1 / 3}}{V^{2 / 3}} Hleft(frac{V^{1 / 3}(|E|-2 V)}{leftlanglevarepsilon^{2}rightrangle^{2 / 3}}right)$$ where $H$ is yet another smooth function that varies from $H(xrightarrow -infty)=-1/8x$ to $H(0)=0.289$ to $H(xrightarrow infty)=sqrt{x}$.

A second model, studied because of its experimental relevance, is a 1D Bose-Einstein condensate in a disordered speckle field. This is a continuous model that is probably less relevant to the OP, but worth mentioning for completeness. The model definition uses the Gross-Pitaevskii equation for a BEC in a harmonic potential with random speckle:

$$ i hbar partial_{t} psi=left[frac{-hbar^{2}}{2 m} partial_{z}^{2}+V_{mathrm{ho}}(z)+V(z)+g|psi|^{2}-muright] psi $$

with the speckle field $V(x)$ characterized by average intensity $V_R$ and correlation length $sigma_R$. Then, the localization length is estimated (in ref. 3) as:

$$xi^{-1}(k)=gamma(k)=frac{pi m^{2} V_{mathrm{R}}^{2} sigma_{mathrm{R}}}{2 hbar^{4} k^{2}} left(1-k sigma_{mathrm{R}}right) Thetaleft(1-k sigma_{mathrm{R}}right)$$

Note that this model has an effective mobility edge even in 1D, which is due to the finite correlation length of the disorder.

If you are interested in other models, searching for results for the Lyapunov exponent might help you to find more information.

References:

Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices

B. A. Van Tiggelen, "Localization of waves," from Diffuse waves in complex media

Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials