Schrieffer–Wolff transformation

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.[1] 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.,[2] Joaquin Mazdak Luttinger and Walter Kohn used this method in an earlier work about non-periodic k·p perturbation theory.[3] 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 the Anderson impurity model.

Derivation

Consider a quantum system evolving under the time-independent Hamiltonian operator ${\displaystyle H}$ of the form:

$${\displaystyle H=H_{0}+V}$$

where ${\displaystyle H_{0}}$ is a Hamiltonian with known eigenstates ${\displaystyle |m\rangle }$ and corresponding eigenvalues ${\displaystyle E_{m}}$, and where ${\displaystyle V}$ is a small perturbation. Moreover, it is assumed without loss of generality that ${\displaystyle V}$ is purely off-diagonal in the eigenbasis of ${\displaystyle H_{0}}$, i.e., ${\displaystyle \langle m|V|m\rangle =0}$ for all ${\displaystyle m}$. Indeed, this situation can always be arranged by absorbing the diagonal elements of ${\displaystyle V}$ into ${\displaystyle H_{0}}$, thus modifying its eigenvalues to ${\displaystyle E'_{m}=E_{m}+\langle m|V|m\rangle }$. 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 ${\displaystyle V}$. This unitary transformation is conventionally written as:

$${\displaystyle H'=e^{S}He^{-S}}$$

When ${\displaystyle V}$ is small, the generator ${\displaystyle S}$ of the transformation will likewise be small. The transformation can then be expanded in ${\displaystyle S}$ using the Baker-Campbell-Haussdorf formula

$${\displaystyle H'=H+[S,H]+{\frac {1}{2}}[S,[S,H]]+\dots }$$

Here, ${\displaystyle [A,B]}$ is the commutator between operators ${\displaystyle A}$ and ${\displaystyle B}$. In terms of ${\displaystyle H_{0}}$ and ${\displaystyle V}$, the transformation becomes

$${\displaystyle H'=H_{0}+V+[S,H_{0}]+[S,V]+{\frac {1}{2}}[S,[S,H_{0}]]+{\frac {1}{2}}[S,[S,V]]+\dots }$$

The Hamiltonian can be made diagonal to first order in ${\displaystyle V}$ by choosing the generator ${\displaystyle S}$ such that

$${\displaystyle [H_{0},S]=V}$$

This equation always has a definite solution under the assumption that ${\displaystyle V}$ is off-diagonal in the eigenbasis of ${\displaystyle H_{0}}$. Substituting this choice in the previous transformation yields:

$${\displaystyle H'=H_{0}+{\frac {1}{2}}[S,V]+O(V^{3})}$$

This expression is the standard form of the Schrieffer–Wolff transformation. Note that all the operators on the right-hand side are now expressed in a new basis "dressed" by the interaction ${\displaystyle V}$ to first order.

In the general case, the difficult step of the transformation is to find an explicit expression for the generator ${\displaystyle S}$. Once this is done, it is straightforward to compute the Schrieffer-Wolff Hamiltonian by computing the commutator ${\displaystyle [S,V]}$. 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 ${\displaystyle V}$ 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.

References

1. 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)
2. 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.
3. 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.

Further reading

• Phillips, Philip (2012). Advanced Solid State Physics (Second ed.). New York: Cambridge University Press. pp. 109–114. ISBN 978-1-107-49346-9.od