Friday, March 19, 2010

Emergence of a hierarchy

The hierarchy of objects and descriptions associated with theories of electronic properties of solids.
[n.b. the arrows should point up not down! I am graphically challenged]
At the level of quantum chemistry one can describe the electronic states of single (and pairs of) molecules in terms of molecular orbitals which are linear combinations of atomic orbitals. [This is Laughlin and Pines, Theory of Everything!]
Just a few of these orbitals interact significantly with neigbouring units in the solid. Low-lying electronic states can be described in terms of itinerant fermions on a lattice and an effective Hamiltonian such as a Hubbard model. In the Mott insulating phase the electrons are localised on single lattice sites and can described by a Heisenberg spin model. The low-lying excitations of these lattice Hamiltonians may have a natural description in terms of quasi-particles which can be described by a continuum field theory such as a non-linear sigma model.

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