Itinerant spin-flip particle-hole continuum

Spin reversal in exchange-split bands creates an electron-hole excitation with energy ${ϵ}_{k+q}-{ϵ}_k+2M$ for a majority-to-minority flip in the simple uniform model with $M>0$. Its allowed continuum depends on occupied initial and empty final states. At $q=0$ all such allowed transitions have exchange energy $2M$; finite momentum changes the thresholds. A collective ferromagnetic magnon can remain gapless below this continuum when spin rotation is exact. The continuum is not the same excitation as a longitudinal amplitude mode.

The edges of an itinerant spin-flip particle-hole continuum follow from both excitation energy and occupation constraints. Consider a zero-temperature three-dimensional parabolic-band model with majority states occupied for $|\mathbf{k}|\le k_F$ and an entirely empty minority band. For exchange splitting $2M$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$, a majority-to-minority flip carrying wavevector $\mathbf{q}$ hasFor $q>0$, the scalar product is bounded by $\pm k_{F}q$, giving continuum edgesThe extremes occur at opposite points of the occupied Fermi surface along $\mathbf{q}$. At $q=0$ the interval collapses to the single exchange energy $2M$.

The empty-minority assumption requires sufficient splitting; otherwise Pauli exclusion principle blocks some final states and the edges must be recomputed over the allowed region. These are independent particle-hole thresholds, not a calculation of the collective ferromagnetic magnon dispersion. A collective mode below the continuum can therefore have a different low-momentum energy scale.