= Schrieffer-Wolff transformation
{c}
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 $H=H_0+\epsilon V$ as
$$
V_d=PVP+QVQ,\qquad V_{od}=PVQ+QVP.
$$
The first part acts within sectors; the second couples them. For an <anti-Hermitian> generator $S=\epsilon S_1+O(\epsilon^2)$, 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> gives
$$
H_{\rm eff}=PH_0P+\epsilon PVP+\frac{\epsilon^2}{2}P[S_1,V_{od}]P+O(\epsilon^3).
$$
In 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.<Schrieffer-Wolff transformation for quantum many-body systems by Sergey Bravyi and others>[Section 3.2, equations (3.6)–(3.15), block decomposition, generator and second-order effective Hamiltonian]{ref}
For a direct two-level check, take $H_0=\operatorname{diag}(0,\Delta)$, $\Delta>0$, and $V=t|0\rangle\langle1|+t^*|1\rangle\langle0|$. The formula predicts the lower <energy> shift $-\epsilon^2|t|^2/\Delta$. Expanding the exact lower <eigenvalue>, $(\Delta-\sqrt{\Delta^2+4\epsilon^2|t|^2})/2$, 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>.<Schrieffer-Wolff transformation for quantum many-body systems by Sergey Bravyi and others>[Section 1.1, historical Anderson-to-Kondo application]{ref}
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