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Analyticity breaking and Anderson localization in incommensurate lattices. (English) Zbl 0943.82510

Horwitz, L. P. (ed.) et al., Group theoretical methods in physics. Proceedings of the 8th international colloquium, Kiryat Anavim, Israel, March 25-29, 1979. Bristol: Adam Hilger. Ann. Isr. Phys. Soc. 3, 133-164 (1980).
Author’s summary: Recently discovered crystals which are now of great interest, both from the experimental and theoretical point of view, exhibit superimposed periodic modulation of the atomic positions, the wave-vector of which is generally incommensurate with the usual reciprocal lattice wave-vectors. From the theoretical point of view, the problem of finding material structures is very hard when it is set by starting from some assumed knowledge of the atomic interactions. This paper shows, for particular oversimplified nontrivial models in which some usual drastic approximations have been avoided, that the concept of analyticity must be considered to understand such complex structures. After some general remarks, two models are studied. The first is an old model, introduced to study crystal dislocations and epitaxial monolayers on a crystal surface. It is also considered to be a good prototype model exhibiting incommensurate structures. The second model deals with a quantum model determining the electronic structure in an incommensurate lattice. An Anderson localization transition occurs and is in fact a transition by breaking of analyticity. (It should also have some applications in other systems where the incommensurate potential comes from a charge density wave created, for example, by electron-electron or lattice-electron interactions).
For the entire collection see [Zbl 01494166].

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Citations:

Zbl 0459.20004