# What is localization length of eigenvectors?

Apology if this question is not appropriate. I was looking to associate entropy to eigenvectors for some of my work and I found the link http://chaos.if.uj.edu.pl/~karol/pdf/ZK94.pdf . This leads to the concept of localization of eigenvectors as mentioned in https://www.researchgate.net/profile/Luca_Molinari/publication/236159789_Scaling_Properties_of_Band_Random_Matrices_Giulio_Casati_Luca_Molinari_and_Felix_Izrailev_Phys._Rev._Lett._64_1851_(1990)/links/00463516883193f770000000.pdf

The abstract of the second link is: "It is shown on the basis of numerical data that the normalised localisation length of eigenvectors of band random matrices follows a scaling law. The scaling parameteris b2/N, where Ь measures the band-width and N is the size of the matrix."

May I ask what exactly 'localization length of eigenvectors' means. I understand eigenvectors are generally unity length and it is important for direction only. Can anyone please help.

• ...why don't you read the paper that introduces that terminology to find out what it means by it? In general quantum mechanics, "localization length of eigenvectors" doesn't mean anything. Mar 21, 2016 at 21:51
• @ACuriousMind Thank you so much. I never imagined that the idea was introduced in that paper only. Thank you. Mar 21, 2016 at 21:54

In certain disordered physical systems, the eigenstates have a localized behavior, in the sense that they fall off exponentially in space like $\psi(x) \approx e^{-x/\xi}$, with $\xi$ defined as the localization length. This is called Anderson localization. If you search for that term here, you can find more information about it, see for example here or here.