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Picturing the Propagation of Localized Electrons*
In some branches of theoretical physics it often turns out that a picture can be worth a thousand quite technical words (or nowadays 1024). These pictures, or 'diagrams', provide a compact, intuitive, and often highly physical representation of objects that arise in theories developed to describe systems of many, many particles, usually interacting. In the ensuing quantum field theory and in perturbation many-body theory, the scattering of (quasi) particles is represented as a convergent series of so-called Feynman diagrams.** Here the probability amplitude for a particle to travel between two scattering events, or space-time points, is given by the so-called Feynman propagator.
The traditional domain of many-body perturbation theory has been spatially uniform systems. What is now presented is a diagrammatic quantum field formalism for calculating quantum mechanical quantities in just the opposite limit. This is a limit where the physics of the system deems the electrons to be strongly localized but where the wave functions still overlap, just a bit. Despite these conceptual differences, a diagrammatic language can be introduced for this important case by direct analogy with that of standard field theory. It is based, in particular, on the idea that overlaps between localized single-electron wave functions can themselves be viewed as quantum perturbations, and these introduce scattering of the single-electron amplitudes. A key quantity in the emerging diagrammatic language is indeed the overlap between single-electron wave functions, and it is measured by a quantity referred to as the overlap integral, which in our formalism becomes the equivalent of the Feynman propagator in the traditional diagrammatic field theory. The resulting framework then turns out to be exceedingly general and flexible, and the corresponding 'rules' for constructing the diagrams for physical objects of interest are quite straightforward. The new diagrammatic method should be of some utility both in theoretical condensed matter physics and in quantum chemistry.
*S.A.Bonev and N.W.Ashcroft, Physical Review B 78, 195117 (2008) [Cornell University]
**R.P.Feynman, Physical Review 76, 769 (1949) [Cornell University]
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