Brief Explanation and Summary of Anderson Localization in Three Dimensional Systems

Fig. 1: Diagram for the expression of Vc(0) for a specific path 0 → 1 → 2 → 3 → 0. The blue solid path indicates the term with no repeated site indices. The purple dashed path indicates one path with repeated indices, 0 → 1 → 2 → 4 → 5 → 6 → 2 → 3 → 0, whose effect can be included in the denominator of site 2, e2, in the path 0 → 1 → 2 → 3 → 0.

Anderson localization refers to the statement that, for a disordered system, transport will not happen for some conditions. In three dimensions, there exists a critical disorder bandwidth such that if the random disorder is stronger than the critical value, transport will not be able to happen and the system will become insulating. In dimensions d ≤ 2, localization will always happen for arbitrary weak disorder, demonstrated by a scaling theory. [1] This is a significant finding providing insights into how transport properties in real materials can be affected by defects or impurity. In this report, we briefly explain and summarize the reasoning of the well-known original work on Anderson localization for a three dimensional system. [2]

The disordered model can be described by the Hamiltonian

where E_j is the random disorder energy on site j within a bandwidth W; V_jk is the interaction between j and k, taken to be constant V over all nearest neighbour correlations and 0 for others. C_j and C^†_j are the Fermion operators at site j. Assuming one spin is put initially on site 0 at t=0, Anderson investigates the time evolution of the probability amplitude of a spin on site 0 at t → +∞. In the end he concluded that as long as the series

converges, the wave functions are localized, which means the probability amplitude at site 0 will not decay to 0 at t → +∞, but will approach a constant. In this expression, s is an arbitrary complex number with positive or zero real part. i is the imaginary unit. Here, the sum is taken over all the possible paths starting at site 0 and ending at site 0, including paths with repeated site indices. However, we can eliminate all of the repeated site indices by including them in the energy denominator e_j=is-E_j-V_c(j) (Fig. 1). We define each term in the summation as T_L. Defining average number of terms of length L between T_L and T_L + dT_L as n(T_L)dT_L, Anderson comes up with n(T) of the following general form considering different cases:

where L(T) is a slowly varying function relative to T. κ is the connectivity in percolation theory so the number of nonrepeating paths of length L leading from any given atom is ∼ κ^L. In the end, Anderson defined a critical value (W/V)₀ satisfying

By establishing a probability distribution for the terms in the series, he concluded that if (W/V) ≳ (W/V)₀, the transport will not happen, which is known as the Anderson localization condition for three-dimensional systems.

© Wen Wang. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] E. Abrahams et al., "Scaling Theory of Localozation: Absence of Quantum Diffusionin Two Dimensions," Phys. Rev. Lett. 42, 673 (1979).

[2] P. W. Anderson, "Absence of Diffusion in Certain Random Lattices," Phys. Rev. 109, 1492 (1958).