Hamilton transformation

We discuss the basic concept of the Schrieffer-Wolff transformation in 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 $U$. In practice, finding the particular unitary transformation $U$ 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 $U$, or the generator $S$ 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 $\mathcal{H}$ that share the a given choice of characteristics, e.g. the local particle quantum number $n_i$. We will denote the set of terms in the Hamiltonian that are already block-diagonal in the initial basis as $H_0$. The remaining terms connect different blocks, i.e are block-offdiagonal, and are denoted $V$. The complete Hamiltonian thus reads $$H=H_0+V.$$ A unitary similarity transformation of the Hamiltonian is given by $$H=U^{†}HU={\text{e}}^{-S}\left(H_0+V\right){\text{e}}^S={\text{e}}^{-S}{H}_0{\text{e}}^S+{\text{e}}^{-S}V{\text{e}}^S,$$ where $S$ 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 $S$ such that the transformed Hamiltonian $H$ becomes entirely block-diagonal. In order to arrive at an equation for $S$ one makes use of the Campbell-Baker-Hausdorff formula to expand $$H={\mathrm{e}}^{S}H{\mathrm{e}}^{-S}=\sum _{m=0}^{\infty }\frac{1}{m!}[S,H{]}_m=H+[S,H]+\frac{1}{2}\left[S,[S,H]\right]+\dots ,$$ where for generators $S$ satisfying $\Vert S\Vert \ll \Vert H_0\Vert$, with $\Vert \cdot \Vert$ a suitable norm, one can approximate the expression as $$H=H_0+V+\left[S,H_0\right]+\left[S,V\right]+\mathcal{O}(S^2).$$ 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 $$[S,H_0]=-V\iff [H_0,S]=V,$$ by which one can determine the generator $S$ which removes the block-offdiagonal terms $V$ of the Hamiltonian. If a solution $S$ exists, one can use $$\left[S,\left[S,H_0\right]\right]=\left[S,-V\right]=-\left[S,V\right]\notin \mathcal{O}(S^2),$$ to simplify the expression for the transformed Hamiltonian $$H={\mathrm{e}}^{-S}\left(H_0+V\right){\mathrm{e}}^S\simeq H_0+\frac{1}{2}\left[S,V\right]+\mathcal{O}(S^2),$$ where the terms originating from $[S,V]/2$ contain the perturbative corrections arising from the consecutive application of two block-offdiagonal operators. A subsequent projection to the subspaces $${\mathcal{P}}_n=\sum _{p\in {\mathcal{P}}_n}|p⟩⟨p|,$$ yields the block-diagonal Hamiltonian $$H_{\mathrm{block}-\mathrm{diagonal}}=\sum _n{\mathcal{P}}_n\tilde{H}{\mathcal{P}}_n,$$ where $n$ denotes the distinct blocks of the Hilbert space.

Symmetry specification of block-offdiagonal operators

The block-offdiagonal part of the Hamiltonian $V$ consists of a sum of block-offdiagonal terms. These in turn comprise products of individually block-offdiagonal operators $x$. In the following we detail a method to decompose generic block-offdiagonal operators $x$ into distinct components. Each of the components exclusively connects two distinct blocks of the Hilbert space $\mathcal{H}$, often associated with distinct quantum numbers of a symmetry of the system. A given block-offdiagonal operator $x:\mathcal{H}\to \mathcal{H}$ satisfies $$\left[H_0,x\right]=\epsilon z\ne 0,$$ where $\epsilon$ denotes an arbitrary scalar and $z:\mathcal{H}\to \mathcal{H}$ an arbitrary operator. Let $A$ 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 $A$ to expand the operator $x$ as $$x=\sum _q{x}_q=\sum _q{\beta }_q\prod _{i\ne q}\left(A-a_i\right)x,$$ where the different $x_q$ couple the target subspace associated with the eigenvalue $a_q$ of $A$ to other subspaces of the Hilbert space. If the operator $A$ satisfies $$[x,A]=0,$$ then both subspaces, initial and final, coupled via $x_q$ are specified by the eigenvalue $a_q$. This is possible for the fermionic creation and annihilation operators $c_{i\sigma }^{†}$ and $c_{i\sigma }$. The coefficients ${\beta }_q$ are solutions to the equation $${\beta }_q\prod _{i\ne q}\left(a_q-a_i\right)=1.$$ The symmetry-specified block-offdiagonal operators $x_q$ satisfy $$\left[x_q,x_{q^{\prime }}^{†}\right]\propto {\delta }_{q{q}^{\prime }},$$ where the operator $x_q^{†}$ denotes the hermitian conjugate of the operator $x_q$.

Schrieffer-Wolff transformation as a system of linear equations for unique block-offdiagonal operators

Following the procedure outlined in the section Perturbative similarity transformation we separate the Hamiltonian of the system into its block-diagonal and block-offdiagonal contributions as $$H=H_0+V.$$ The block-offdiagonal contribution $V$ comprises each block-offdiagonal term $v$ $$\begin{gathered} V=\sum _{\left\{v\right\}}{\alpha }_v\left[\left(\prod _j{o}^j\prod _i{x}^i\right)+{\left(\prod _j{o}^j\prod _i{x}^i\right)}^{†}\right] \\ =\sum _{\left\{v\right\}}{\alpha }_v\left[\left(\prod _{i,j}{o}^j\sum _{q(i)}{x}_{q(i)}^i\right)+{\left(\prod _{i,j}{o}^j\sum _{q(i)}{x}_{q(i)}^i\right)}^{†}\right], \end{gathered}$$ where ${\prod }_i{x}^i$denotes sequences of individually block-offdiagonal operators, ${\prod }_j{o}^j$ denotes sequences of individually block-diagonal operators, and $x_{q(i)}^i$ denotes the symmetry-specified components of the operator $x^i$. We introduce a vector spaces ${\mathcal{V}}_0^h$ and ${\mathcal{V}}_0^a$, for which each unique pair of hermitian, or anti-hermitian respectively, symmetry-specified operator sequences in $V$ corresponds to a unique orthonormal basis vector $${\hat{e}}_v=\{\begin{array}{ll}\left({\prod }_j{o}^j{\prod }_i{x}_{q(i)}^i\right)+{\left({\prod }_j{o}^j{\prod }_i{x}_{q(i)}^i\right)}^{†}& \in {\mathcal{V}}_0^h\\ \left({\prod }_j{o}^j{\prod }_i{x}_{q(i)}^i\right)-{\left({\prod }_j{o}^j{\prod }_i{x}_{q(i)}^i\right)}^{†}& \in {\mathcal{V}}_0^a\end{array},$$ where ${\mathcal{V}}^h$ denotes the hermitian vector space and ${\mathcal{V}}^a$ the anti-hermitian vector space. We define the linear map $$L:\begin{array}{l}{\mathcal{V}}_0^h\to {\mathcal{V}}_1^a\\ {\mathcal{V}}_0^a\to {\mathcal{V}}_1^h\end{array},$$ where the action of the linear map on a vector ${\hat{e}}_v$ is given by $$L{\hat{e}}_v=\left[H_0,{\hat{e}}_v\right],$$ where $$\begin{array}{lcl}(L{\hat{e}}_v)\in {\mathcal{V}}_1^a& \mathrm{if}& {\hat{e}}_v\in V_0^h\\ (L{\hat{e}}_v)\in {\mathcal{V}}_1^h& \mathrm{if}& {\hat{e}}_v\in V_0^a\end{array},$$ with $$\dim \left({\mathcal{V}}_1\right)\ge \dim \left({\mathcal{V}}_0\right),$$ since ${\mathcal{V}}_1$ includes additional unique operator sequences generated by $[H_0,{\hat{e}}_v]$. In the vector spaces $\mathcal{V}$, we can translate the equation for the generator $S$ of the Schrieffer-Wolff transformation to $$L\vec{S}=\vec{V}\iff \left[H_0,S\right]=V,$$ with $\vec{S}\in {\mathcal{V}}_0^a$ and $\vec{V}\in {\mathcal{V}}_1^h$. Determining the generator $S$ of the Schrieffer-Wolff transformation becomes equivalent to solving the set of linear equations. In general one finds $$\mathrm{rank}(L)\le \dim \left({\mathcal{V}}_0\right)\le \dim \left({\mathcal{V}}_1\right),$$ and there is consequently no unique solution $\vec{S}$ to the set of equations. It is intuitive that the terms of $V$ approximately bring about transitions between distinct eigenstates of $H_0$ belonging to different blocks of the Hilbert space. This is reflected by $$\left[H_0,{\hat{e}}_v^h\right]=\left(\Delta E_{0,v}\prod _l{o}^l\right){\hat{e}}_v^a,$$ where $\Delta E_{0,v}$ denotes the eigenvalue difference between the two eigenstates of $H_0$ and ${\prod }_l{o}^l$ denotes an arbitrary sequence of individually block-diagonal operators. From the previous equations we can approximate $$\Vert \vec{S}\Vert \approx \sqrt{\sum _{\{v\}}{\left(\frac{{\alpha }_v}{\Delta E_{0,v}}\right)}^2},$$ which highlights the necessity for a significant energy gap $\Delta E_{0,v}$ between the separate subspaces of the Hilbert space coupled by $V$ in order for the series expansion of $H$ to $\mathcal{O}(S^2)$ to be considered a good approximation. We consider terms $v$ for which $$\frac{{\alpha }_v^2}{|\Delta E_{0,v}|}\ge 1,$$ to be the resonant terms of $V$ which should be retained in the transformed Hamiltonian $H$. To identify these resonant terms of $V$, we employ a singular value decomposition (SVD) of the linear map $L$ and arrive at $$L=U\Sigma W^{†}=U\left({\Sigma }_{>}+{\Sigma }_{<}\right)W^{†}=L_{\mathrm{gapped}}+L_{\mathrm{resonant}},$$ where ${\Sigma }_{>}$ comprises the singular values $${\sigma }_i>\frac{1}{N_v}\sum _{\{v\}}\sqrt{{\alpha }_v^2},$$ where $N_v$ denotes the number of terms $v$. We define the Moore-Penrose pseudoinverse of $L$ that acts exclusively on the gapped, i.e. non-resonant, terms in $V$ as $$L_{\mathrm{gapped}}^{+}=\left(W{\Sigma }_{>}^{+}{U}^{†}\right),$$ where ${\Sigma }_{>}^{+}$ denotes the pseudoinverse of ${\Sigma }_{>}$. In the absence of a unique solution to $\left[H_0,S\right]=V$, the closest approximate solution is given by $$\vec{S}=\left(L_{\mathrm{gapped}}^{+}\vec{V}\right)\in {\mathcal{V}}_0^a,$$ and the resulting transformed Hamiltonian reads $$\tilde{H}=H_0+V_{\mathrm{resonant}}+\left[S,V_{\mathrm{resonant}}\right]+\frac{1}{2}\left[S,V_{\mathrm{gapped}}\right],$$ where $V_{\mathrm{gapped}}=[H_0,S]$ and $V_{\mathrm{resonant}}=(V-V_{\mathrm{gapped}})$.