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For a unitary transformation $e^S$, take $S^{†}=-S$ and $S=O(t)$. Operator conjugation by an exponential givesSeparate the unperturbed energy $\mathcal{E}(\Delta )$ from $H_t=O(t)$. To first order,Therefore eliminating the first-order hopping requiresThis is a Schrieffer-Wolff transformation: virtual transitions are removed from the leading Hamiltonian and reappear as higher-order effects within a retained occupation sector. The same canonical-elimination strategy relates localized-moment and effective-exchange descriptions in the original construction.^[ref]

Such a choice requires the hopping to connect separated unperturbed energies. If it connects exactly degenerate states, those states must remain in the retained degenerate block; a singular perturbative generator is not an acceptable replacement.

A theoretical condensed matter physicist who coauthored the Schrieffer-Wolff transformation with John Robert Schrieffer. Their canonical transformation relates a model with fluctuating impurity charge to an effective model of spin exchange in the weak-mixing limit. This explains how virtual charge processes can produce the low-energy physics associated with the Kondo effect.^[ref]

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]

Sergey Bravyi developed rigorous treatments of the Schrieffer-Wolff transformation with David DiVincenzo and Daniel Loss. Their 2011 work derives low-energy effective quantum dynamics through a unitary transformation that decouples low- and high-energy subspaces.^[ref]

For a Quantum Hamiltonian $H$, an anti-Hermitian generator $S$ gives $H_{\mathrm{eff}}=P{e}^{S}H{e}^{-S}P$ with $P$ the orthogonal projection onto the chosen low-energy subspace, when the transformation is constructed appropriately. Bounding the approximation matters: formally discarding high-energy states without controlling their virtual coupling can miss low-energy corrections.