A real perturbation with zero diagonal and every off-diagonal element $t$ has one eigenvalue $2t$ on the symmetric vector and a twofold eigenvalue $-t$ on its orthogonal complement. It is $t(J_3-I)$, where $J_3$ is the all-ones matrix. Diagonal entries alone do not give first-order energies in a degenerate subspace.

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]

The number of quantum states assigned to an energy level or grouped spectroscopic level. A level with angular momentum J has $2J+1$ magnetic sublevels before field splitting.

In a mean-field diffusion model, an effective inverse-area measure for removal of mobile defects. The loss rate $D{k}^{2}c$ has number-density-per-time units when $D$ is diffusivity and $c$ is defect number density.

Preferential absorption of self-interstitials relative to vacancies by a sink. Dislocation bias contributes to the unequal defect fluxes that can sustain void swelling.

A binary De Bruijn sequence of order $n\ge 1$ is a cyclic binary string in which every length-$n$ binary string occurs exactly once as a consecutive window. It has length $2^n$. For $n=3$, 00010111 has cyclic windows 000, 001, 010, 101, 011, 111, 110, 100; the windows at the end wrap around to the beginning. A linear recording must repeat the first $n-1$ symbols at its end. Finding an Eulerian cycle in the De Bruijn graph on length-$(n-1)$ words constructs one: each edge supplies exactly one required window.

A matter component with relativistic motion during an important part of structure formation. Its free streaming suppresses small-scale perturbations. Making a light thermal neutrino Majorana rather than Dirac does not by itself turn this component into cold dark matter.