Magnon-phonon interaction and the underlying role of the Pauli exclusion principle

The magnon-phonon interaction is receiving growing attention due to its key role in spin caloritronics and the emerging field of acoustic spintronics. At resonance, this magnetoelastic interaction forms magnon polarons, which underpin exotic phenomena such as magnonic heat currents and phononic spin, but is mostly investigated using mesoscopic spin-lattice models. Motivated to integrate the magnon-phonon interaction into first-principles many-body electronic structure theory, we set out to derive the exchange contribution, which is subtler than the spin-orbit contribution, using Schwinger functional derivatives. To avoid having to solve the famous Hedin-Baym equations self-consistently, the phonons are treated as perturbations to the electronic structure. A formalism based on imposing a crossing-symmetric electron-electron interaction is developed in order to treat charge and spin on equal footing to respect the Pauli exclusion principle. Due to spin conservation, the magnon-phonon interaction first enters to second order through the magnon-magnon interaction, which renormalizes the magnons. We show by iteration that the magnon-magnon interaction contains a “screened $T$ matrix” term and an arguably more important term which, in the local-spin limit, enables first-principles phonon emission and absorption amplitudes, predicted by phenomenological magnetoelastic models. These terms are, respectively, of first and second order in the screened collective four-point interaction $\mathcal{W}$—a crossing-symmetric analog of Hedin's $W$. Proof-of-principle results are presented at varying temperatures for an isotropic magnon spectrum in three dimensions in the presence of a flat optical phonon branch.

Figure 1

Diagrammatic representation of the part of the spin-flip interaction describing simultaneous propagation of spin and charge fluctuations. Spin $\uparrow$ and $\downarrow$ Green's functions are distinguished by their colors, and spin conservation in ${{\mathcal{W}}^{·}}_{\downarrow \uparrow }^{\downarrow \uparrow }$ is emphasized with the same colors.

Figure 2

Real and imaginary parts of the PAMM interaction ${{\mathrm{\Delta }}_{\text{PA}}^r}_{\downarrow \uparrow }^{\downarrow \uparrow }\left(\omega \right)$ at $T=0K$ and $T=300K$, with $\mathcal{A}=32meV$.

Figure 3

Magnitude of the renormalized magnon spectral function $A^{\text{M}}(k,\omega )$ at $T=0K$ and $T=300K$ for various $\mathcal{A}$.

Figure 4

Magnitude of the total spectral function $A^{\text{M}}\left(\omega \right)$ for Fock magnons ${r^r}_{\downarrow \uparrow }^{\downarrow \uparrow }$ and renormalized magnons ${{\mathcal{R}}^r}_{\downarrow \uparrow }^{\downarrow \uparrow }$ at $T=0K$ and $T=300K$ for various $\mathcal{A}$.

Figure 5

Product of Bose occupation $n_B^{\beta }\left(\omega \right)$ and magnon spectrum $A^{\text{M}}\left(\omega \right)$ for Fock magnons ${r^r}_{\downarrow \uparrow }^{\downarrow \uparrow }$ and renormalized magnons ${{\mathcal{R}}^r}_{\downarrow \uparrow }^{\downarrow \uparrow }$ at various temperatures. $\mathcal{A}=64meV$.

Figure 6

Spin magnetization $m$ as a function of $T$ for different values of $\mathcal{A}$, together with the associated Curie temperatures $T_c$ for which $m=0$.