Non-Landau Damping of Magnetic Excitations in Systems with Localized and Itinerant Electrons

We discuss the form of the damping of magnetic excitations in a metal near a ferromagnetic instability. The paramagnon theory predicts that the damping term should have the form $\gamma (q,\mathrm{\Omega })\propto \mathrm{\Omega }/\mathrm{\Gamma }(q)$, with $\mathrm{\Gamma }(q)\propto q$ (the Landau damping). However, the experiments on uranium metallic compounds $\mathrm{UGe}{}_2$ and UCoGe showed that $\mathrm{\Gamma }(q)$ is essentially independent of $q$. A nonzero $\gamma (q=0,\mathrm{\Omega })$ is impossible in systems with one type of carrier (either localized or itinerant) because it would violate the spin conservation. It has been conjectured recently that a near-constant $\mathrm{\Gamma }(q)$ in $\mathrm{UGe}{}_2$ and UCoGe may be due to the presence of both localized and itinerant electrons in these materials, with ferromagnetism involving predominantly localized spins. We present the microscopic analysis of the damping of near-critical localized excitations due to interaction with itinerant carriers. We show explicitly how the presence of two types of electrons breaks the cancellation between the contributions to $\mathrm{\Gamma }(0)$ from the self-energy and vertex correction insertions into the spin polarization bubble. We compare our theory with the available experimental data.

Figure 1

(a)–(c) The spin-polarization bubble $\chi (q,\mathrm{\Omega })$. The thin lines are fermionic propagators, the wavy line is the effective interaction mediated by the total spin susceptibility ${\chi }^{\text{tot}}(q,\mathrm{\Omega })=\chi (q,\mathrm{\Omega })/[1-\overline{U}\chi (q,\mathrm{\Omega })]$. Each side vertex and each interaction vertex contains spin $\sigma$ matrices (not specified). (a) The bubble for free fermions ${\chi }_0(q,\mathrm{\Omega })$; (b) spin-polarization bubble with self-energy and MT-type vertex corrections. (c) The two AL diagrams. By power counting, they are of higher order, but in fact, they are of the same order as the diagrams in (b) (see text). (d) One-loop diagram for the self-energy of an itinerant fermion in a a two-component model, due to interaction with localized spins. The wavy lines are ${\chi }_L(q,\mathrm{\Omega })$ for localized electrons.

Figure 2

Fitting of the data for $\mathrm{\Gamma }(T){\xi }^2$ from the work of Ref. [1]. The solid line is the fit by the dependence $\propto T^2$. The dashed portion of the line shows the regime around $T_c\simeq 54\mathrm{K}$, where $\mathrm{\Gamma }{\xi }^2\propto {\mathrm{\Sigma }}^{\prime \prime }(T)$ gets an additional enhancement due to the increase of ${\mathrm{\Sigma }}^{\prime \prime }(T)$.