Transformation of the many-body Hamiltonian
We discuss the basic concept of the Schrieffer-Wolff transformation in the section Perturbative similarity transformation. We detail our proposed extended Schrieffer-Wolff transformation method in sections Symmetry specification of block-offdiagonal operators and Schrieffer-Wolff transformation as a system of linear equations for unique block-offdiagonal operators.
Perturbative similarity transformation
In principle, any Hamiltonian can be completely diagonalized by means of a particular unitary transformation . In practice, finding the particular unitary transformation often requires a complete diagonalization of the Hamiltonian to begin with. Here we recap how a perturbative similarity transformation, namely the Schrieffer-Wolff transformation, can be used to determine an approximate transformation operator , or the generator thereof, which does not diagonalize the Hamiltonian fully, but yields a block-diagonal Hamiltonian instead. These blocks consist of the orthonormal states in the Hilbert space that share the a given choice of characteristics, e.g. the local particle quantum number . We will denote the set of terms in the Hamiltonian that are already block-diagonal in the initial basis as . The remaining terms connect different blocks, i.e are block-offdiagonal, and are denoted . The complete Hamiltonian thus reads
A unitary similarity transformation of the Hamiltonian is given by
where is an anti-hermitian operator. One refers to it as the generator of the Schrieffer-Wolff transformation. The key problem of the Schrieffer-Wolff transformation is finding the generator such that the transformed Hamiltonian becomes entirely block-diagonal. In order to arrive at an equation for one makes use of the Campbell-Baker-Hausdorff formula to expand
where for generators satisfying , with a suitable norm, one can approximate the expression as
Considering that commutators of pairs of block-diagonal operators or pairs of block-offdiagonal operators respectively generally become block-diagonal, while the commutators of block-diagonal operators with block-offdiagonal operators become block-offdiagonal, one chooses the equation
by which one can determine the generator which removes the block-offdiagonal terms of the Hamiltonian. If a solution exists, one can use
to simplify the expression for the transformed Hamiltonian
where the terms originating from contain the perturbative corrections arising from the consecutive application of two block-offdiagonal operators. A subsequent projection to the subspaces
yields the block-diagonal Hamiltonian
where denotes the distinct blocks of the Hilbert space.
Symmetry specification of block-offdiagonal operators
The block-offdiagonal part of the Hamiltonian consists of a sum of block-offdiagonal terms. These in turn comprise products of individually block-offdiagonal operators . In the following we detail a method to decompose generic block-offdiagonal operators into distinct components. Each of the components exclusively connects two distinct blocks of the Hilbert space , often associated with distinct quantum numbers of a symmetry of the system. A given block-offdiagonal operator satisfies
where denotes an arbitrary scalar and an arbitrary operator. Let be a diagonal operator in the initial basis. It can be identical to the symmetry operator differentiating the blocks of the Hilbert space, but it is not required to be. One can use the spectrum of to expand the operator as
where the different couple the target subspace associated with the eigenvalue of to other subspaces of the Hilbert space. If the operator satisfies then both subspaces, initial and final, coupled via are specified by the eigenvalue . This is possible for the fermionic creation and annihilation operators and . The coefficients are solutions to the equation The symmetry-specified block-offdiagonal operators satisfy where the operator denotes the hermitian conjugate of the operator .