Schrieffer-Wolff transformation

The Schrieffer-Wolff transformation is a unitary conjugation that decouples selected low- and high-energy sectors to construct an effective Hamiltonian. It is a form of degenerate perturbation theory: states within the retained sector are treated together.

Let an orthogonal projection $P$ commute with $H_0$, put $Q=I-P$, and decompose the perturbation in $H=H_0+ϵV$ asThe first part acts within sectors; the second couples them. For an anti-Hermitian generator $S=ϵS_1+O({ϵ}^2)$, choose $[S_1,H_0]=-V_{od}$. It is the inter-sector coupling, rather than the whole perturbation, that this commutator removes. Projecting the consistently transformed Hamiltonian givesIn an eigenbasis of $H_0$, cross-sector entries obey $(S_1{)}_{mn}=V_{mn}/(E_m-E_n)$. A nonzero separating spectral gap controls these denominators; resonant coupled states belong in a common retained block.^[ref]

For a direct two-level check, take $H_0=\mathrm{diag}(0,\Delta )$, $\Delta >0$, and $V=t|0⟩⟨1|+t^{\ast }|1⟩⟨0|$. The formula predicts the lower energy shift $-{ϵ}^2|t{|}^2/\Delta$. Expanding the exact lower eigenvalue, $(\Delta -\sqrt{{\Delta }^2+4{ϵ}^2|t{|}^2})/2$, gives the same result. A diagonal perturbation would also contribute at first order through $PVP$.

John Robert Schrieffer and P. A. Wolff used the method to relate impurity models underlying the Kondo effect.^[ref]

A perturbative generator dividing by an unperturbed energy difference cannot eliminate a nonzero coupling between exactly degenerate states. Those states must remain in the retained degenerate block. Small denominators also invalidate an expansion when the actual hopping matrix element, including occupation enhancement, is not small in comparison.

If site-transfer energies are $\mathcal{E}(\Delta )$, the left-ordered raising-hop generator has coefficient $f(\Delta )=t/[\mathcal{E}(\Delta )-\mathcal{E}(\Delta -2)]$. Nonlinear occupation energies generally make the effective second-order shifts and spin-exchange amplitudes depend on the occupation sector.

With $X={\sum }_s{a}_s^{†}{b}_s$, $Y=X^{†}$ and a real coefficient $f$, the fixed-occupancy second-order operator is $t[f(\Delta )XY-f(\Delta +2)YX]$. Normal ordering gives $XY=N_a+P$ and $YX=N_b+P$, where $P={\sum }_{s{s}^{\prime }}{a}_s^{†}{a}_{s^{\prime }}{b}_{s^{\prime }}^{†}{b}_s$ exchanges internal states without changing site totals.