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| Fig. 1: Illustration of a typical eigenstate localized due to disorder. (Source: J. San Miguel, after Yoshino et al. [2]) |
In "Absence of Diffusion in Certain Random Lattices", Anderson tackles the problem of non-interacting atoms on a lattice, living on sites i, j, with a Hamiltonian given by [1]
where Ek are random onsite energies lying in a range [-W,W], and Vij are coupling terms between sites. In the infinite-disorder limit, we expect the eigenstates of this Hamiltonian to be perfectly localized. In the zero-disorder limit, meanwhile, the eigenstates are eigenstates of momentum, and states are delocalized. In the latter case, a state localized at t=0 will diffuse throughout the system at large times. It is natural to expect, then, that a finite amount of disorder can limit diffusion, but how does this process occur?
Anderson shows that, under the right conditions, there is a critical coupling scale V0 below which disorder turns off diffusion completely. In other words, even at infinite times and infinite system sizes, there is a finite probability of finding an atom at the site that it originally started on. This occurs because the exact eigenstates of the system are nearly localized on each site, with amplitudes falling off exponentially at long distances (Fig. 1). This disorder-induced transition can be shown to exist as long as the coupling falls off with distance faster than r-3.
This transition has many important implications. First, it can lead to a metal-insulator transition in systems of electrons. Second, it also presents a mechanism by which thermalization can fail to occur. This localization even plays a role in creating conductivity plateaus in the quantum Hall effect.
In this review, we outline Anderson's original derivation of localization in non-interacting systems. The central idea of this derivation is perturbation theory in the number of hoppings between lattice sites. We also briefly discuss the significant body of work following Anderson's original paper. This work includes the scaling theory of localization and many-body localization, as well as many numerical and experimental studies.
© Jonathan San Miguel. 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.
[1] P. W. Anderson, "Absence of Diffusion in Certain Random Lattices," Phys. Rev. 109, 1492 (1958).
[2] S. Yoshino and M. Okazaki, "Numerical Study of Electron Localization in Anderson Model for Disordered Systems: Spatial Extension of Wavefunction, " J. Phys. Soc. Jpn. 43, 415 (1977).