Help - Anderson localization

[weiss_tal]

Hello everyone,

I'm a graduate student and I am doing a simple work on 1D Anderson localization. I need to derive the expression for the localization length when the energies are randomly distributed in the region of $$\left[ \frac{W}{2},-\frac{W}{2} \right]$$. I know the localization length in the limit $$W<<V$$, where $$V$$ are the of diagonal elements of the hamiltonian, is $$\frac{W^2}{96V^2}$$. this expression can be derived while using the second order green function treatment. Since quantum mechanics is not my main studies, I didn't understand the derivation from the green function (I have only found it in Thouless book - Ill condensed matter). If someone know how to derive it, It will help me a lot, or at least know a good reference which explains it simply.
forgive me for my bad English.
thanks,
Tal Weiss.

 

[eljose79]

Dear Tal Weiss,

Thank you for sharing your work on 1D Anderson localization. It's an interesting topic and I'm glad to see that you are delving into it as a graduate student.

To derive the expression for the localization length in the case of randomly distributed energies, we can use the second order green function treatment. This approach involves calculating the average of the square of the green function, which is given by:

\langle G^2(E) \rangle = \frac{1}{N} \sum_{i=1}^N \frac{1}{(E-E_i)^2}

where N is the number of sites and E_i are the random energies in the range of \left[ \frac{W}{2},-\frac{W}{2} \right]. In the limit of W<<V, where V are the off-diagonal elements of the Hamiltonian, we can approximate the average as:

\langle G^2(E) \rangle \approx \frac{1}{N} \sum_{i=1}^N \frac{1}{E_i^2}

Using this approximation, we can then calculate the localization length as:

\xi = \frac{1}{2} \frac{d \ln \langle G^2(E) \rangle}{dE} \Bigg|_{E=0}

Substituting the average expression and solving for the derivative, we get:

\xi = \frac{W^2}{96V^2}

I hope this explanation helps in understanding the derivation. As for a good reference, you can refer to the book "Introduction to Quantum Mechanics" by David J. Griffiths, which has a detailed explanation of the second order green function treatment. If you have any further questions, please feel free to reach out.