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In quantum mechanics, the Schrieffer–Wolff transformation is a unitary transformation used to perturbatively diagonalize the system Hamiltonian to first order in the interaction. As such, the Schrieffer-Wolff transformation is an operator version of second-order perturbation theory. The Schrieffer-Wolff transformation is often used to project out the high energy excitations of a given quantum many-body Hamiltonian in order to obtain an effective low energy model. The Schrieffer–Wolff transformation thus provides a controlled perturbative way to study the strong coupling regime of quantum-many body Hamiltonians.
Although commonly attributed to the paper in which Kondo model was obtained from the Anderson impurity model by J.R. Schrieffer and P.A. Wolff., Joaquin Mazdak Luttinger and Walter Kohn used this method in an earlier work about non-periodic k·p perturbation theory . Using the Schrieffer–Wolff transformation, the high energy charge excitations present in Anderson impurity model are projected out and a low energy effective Hamiltonian is obtained which has only virtual charge fluctuations. For the Anderson impurity model case, the Schrieffer–Wolff transformation showed that Kondo model lies in the strong coupling regime of Anderson impurity model.
Consider a quantum system evolving under the time-independent Hamiltonian operator of the form:
The Schrieffer-Wolff transformation is a unitary transformation which expresses the Hamiltonian in a basis (the "dressed" basis) where it is diagonal to first order in the perturbation . This unitary transformation is conventionally written as:
In the general case, the difficult step of the transformation is to find an explicit expression for the generator . Once this is done, it is straightforward to compute the Schrieffer-Wolff Hamiltonian by computing the commutator . The Hamiltonian can then be projected on any subspace of interest to obtain an effective projected Hamiltonian for that subspace. In order for the transformation to be accurate, the eliminated subspaces must be energetically well separated from the subspace of interest, meaning that the strength of the interaction must be much smaller than the energy difference between the subspaces. This is the same regime of validity as in standard second-order perturbation theory.
- Bravyi, S., DiVincenzo, D. and Loss, D. (2011). "Schrieffer-Wolff transformation for quantum many-body systems". Annals of Physics. 326 (10): 2793–2826. arXiv:1105.0675. Bibcode:2011AnPhy.326.2793B. doi:10.1016/j.aop.2011.06.004.CS1 maint: Multiple names: authors list (link)
- Schrieffer, J.R.; Wolff, P.A. (September 1966). "Relation between the Anderson and Kondo Hamiltonians". Physical Review. 149 (2): 491–492. Bibcode:1966PhRv..149..491S. doi:10.1103/PhysRev.149.491.
- Luttinger, J.R.; Kohn, P.A. (February 1955). "Motion of Electrons and Holes in Perturbed Periodic Fields". Physical Review. 97 (4): 869–883. Bibcode:1955PhRv...97..869L. doi:10.1103/PhysRev.97.869.