It is hard to overstate the impact that Anderson had on the field. In terms of pure scientific results, there are others far more skilled than I who can describe his contributions, but I will mention a few that are well known:
- He developed what is now known as the Anderson model, a theoretical treatment originally intended to capture the essential physics in some transition metal-based magnets. The model considers comparatively localized d orbitals and includes both hopping to neighboring sites in a lattice as well as the "on-site repulsion" U that makes it energetically expensive to have two electrons (in a spin singlet) on the same site. This leads to "superexchange" processes, where energetically costly double-occupancy is a virtual intermediate state. The Anderson model became the basis for many developments - allow coupling between the local sites and delocalized s or p bands, and you get the Kondo model. Put in coupling to lattice vibrations and you get the Anderson-Holstein model. Have a lattice and make the on-site repulsion really strong, and you get the Hubbard model famed in correlated electron circles and as the favored treatment of the copper oxide superconductors.
- Anderson also made defining contributions to the theory of localization. Electrons in solids are wavelike, and in perfect crystal lattices the ones in the conduction and valence bands propagate right past the ions because the waves themselves account for the periodicity of the lattice. Anderson showed that even in the absence of interactions (the electron-electron repulsion), disorder can scatter those waves, and interference effects can lead to situations where the final result is waves that are exponentially damped with distance. This is called Anderson localization, and it applies to light and sound as well as electrons. With strict conditions, this result implies that (ignoring interactions) infinitesimal amounts of disorder can make a 2D electronic system an insulator.
- Here is his Nobel Lecture, by the way, that really focuses on these two topics.
- In considering superconductivity, Anderson also discovered what is now known as the Higgs mechanism, showing that while the bare excitations of some quantum field theory could be massless, coupling those excitations to some scalar field whose particular value broke an underlying symmetry could lead to an effective mass term (in the sense of how momentum and energy relate to each other) for the originally massless degrees of freedom. Since Anderson himself wrote about this within the last five years, I have nothing to add.
- Anderson also worked on superfluidity in 3He, advancing understanding of this first-discovered non-electronic paired superfluid and its funky properties due to p-wave pairing.
- With the discovery of the copper oxide superconductors, Anderson introduced the resonating valence bond (RVB) model that still shapes discussions of these and exotic spin-liquid systems.
Anderson was unquestionably a brilliant person who in many ways defined the modern field of condensed matter physics. He was intellectually active right up to the end, and he will be missed. (For one of my own interactions with him, see here.)