In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the Schrieffer–Wolff transformation is a
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
used to perturbatively
diagonalize the system
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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
A project is any undertaking, carried out individually or collaboratively and possibly involving research or design, that is carefully planned to achieve a particular goal.
An alternative view sees a project managerially as a sequence of even ...
out the high energy excitations of a given quantum many-body
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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 the
Kondo model was obtained from the
Anderson impurity model
The Anderson impurity model, named after Philip Warren Anderson, is a Hamiltonian that is used to describe magnetic impurities embedded in metals.
It is often applied to the description of Kondo effect-type problems, such as heavy fermion sys ...
by
J.R. Schrieffer and P.A. Wolff.,
Joaquin Mazdak Luttinger
Joaquin (Quin) Mazdak Luttinger (December 2, 1923 – April 6, 1997) was an American physicist well known for his contributions to the theory of interacting electrons in one-dimensional metals (the electrons in these metals are said to be in ...
and
Walter Kohn
Walter Kohn (; March 9, 1923 – April 19, 2016) was an Austrian-American theoretical physicist and theoretical chemist.
He was awarded, with John Pople, the Nobel Prize in Chemistry in 1998. The award recognized their contributions to the unde ...
used this method in an earlier work about non-periodic
k·p perturbation theory
In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure (particularly effective mass) and optical properties of crystalline solids.
It is pronounced "k dot p", and is al ...
.
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 the 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
of the form:
where
is a Hamiltonian with known eigenstates
and corresponding eigenvalues
, and where
is a small perturbation. Moreover, it is assumed without loss of generality that
is purely off-diagonal in the eigenbasis of
, i.e.,
for all
. Indeed, this situation can always be arranged by absorbing the diagonal elements of
into
, thus modifying its eigenvalues to
.
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:
When
is small, the generator
of the transformation will likewise be small. The transformation can then be expanded in
using the
Baker-Campbell-Haussdorf formula
Here,_
Here,
Here,