In quantum mechanics, the Schrieffer–Wolff transformation is a unitary transformation used to determine an effective (often low-energy) Hamiltonian by decoupling weakly interacting subspaces.
The transformation also perturbatively diagonalizes the system Hamiltonian to first order in the interaction.
In this, 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.
[1] The Schrieffer–Wolff transformation thus provides a controlled perturbative way to study the strong coupling regime of quantum-many body Hamiltonians.
Wolff.,[3] Joaquin Mazdak Luttinger and Walter Kohn used this method in an earlier work about non-periodic k·p perturbation theory.
[4] 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.
Consider a quantum system evolving under the time-independent Hamiltonian operator
Indeed, this situation can always be arranged by absorbing the diagonal elements of
Substituting this choice in the previous transformation yields:
Note that all the operators on the right-hand side are now expressed in a new basis "dressed" by the interaction
In the general case, the difficult step of the transformation is to find an explicit expression for the generator
This is the same regime of validity as in standard second-order perturbation theory.
This section will illustrate how to practically compute the Schrieffer-Wolff (SW) transformation in the particular case of an unperturbed Hamiltonian that is block-diagonal.
But first, to properly compute anything, it is important to understand what is actually happening during the whole procedure.
being unitary, it does not change the amount of information or the complexity of the Hamiltonian.
The resulting shuffle of the matrix elements creates, however, a hierarchy in the information (e.g. eigenvalues), that can be used afterward for a projection in the relevant sector.
In addition, when the off-diagonal elements coupling the blocks are much smaller than the typical unperturbed energy scales, a perturbative expansion is allowed to simplify the problem.
In physics, and in the original motivation for the SW transformation, it is desired that each block corresponds to a distinct energy scale.
In particular, all degenerate energy levels should belong to the same block.
takes now on a specific meaning: the typical matrix element coupling different sectors must be much smaller than the eigenvalue differences between those sectors.
incorporating ("integrating out") the effects of the other blocks via the perturbation
In the end, it is sufficient to look at the sector of interest (called a projection) and to work with the chosen effective Hamiltonian to compute, for instance, eigenvalues and eigenvectors.
In physics, this would generate effective low- (or high-)energy Hamiltonians.
As mentioned in the previous section, the difficult step is the computation of the generator
To obtain results comparable to second-order perturbation theory, it is enough to solve the equation
becomes very simple, since we obtain an explicit expression, in components:
Using the last formula in the derivation, the second-order Schrieffer-Wolff-transformed Hamiltonian
This is applicable since the SW transformation is based on the approximation
The "dressed" states themselves can be derived, in first-order perturbation theory too, as