Probing hydrodynamic sound modes in magnon fluids using spin magnetometers

The noninteracting magnon gas description in ferromagnets breaks down at finite magnon density where momentum-conserving collisions between magnons become important. Here we present a hydrodynamic description of spin systems with global SU(2) symmetry in the ferromagnetic phase. We identify a key signature of the collision-dominated hydrodynamic regime—a magnon sound mode—which governs dynamics at low frequencies. The magnon sound mode is an excitation of the longitudinal spin component with frequencies below the spin-wave continuum in gapped ferromagnets and can be detected with recently introduced spin qubit magnetometers. We also show that, in the presence of exchange interactions with SU(2) symmetry, the ferromagnet hosts an usual hydrodynamic regime that lacks Galilean symmetry. We show that our results are relevant to ferromagnetic insulators in a finite energy/temperature window such that dipolar and magnon–phonon interactions are negligible, as well as in recent experiments in cold atomic gases.

Figure 1

Spectral function $\chi (q,\omega )={\chi }_{+-}(q,\omega )+{\chi }_{-+}(q,\omega )+4{\chi }_{zz}(q,\omega )$ exhibiting single magnon excitations at the Zeeman energy $\omega =\mathrm{\Delta }$, induced by a finite $\langle S^{-}{S}^{+}\rangle$, and a linearly dispersing sound mode at low frequencies induced by magnon density fluctuations, $\langle {\widehat{S}}^z{\widehat{S}}^z\rangle$. The sound mode is damped above frequencies ${\omega }_{*}$ by viscous forces.

Figure 2

Relaxation time [normalized by $sinh(\omega /2T)]$ of a spin qubit located a distance $d$ from the 2D ferromagnet. Besides the characteristically large relaxation rate induced by spin relaxation due to emission of spin waves at energy $\mathrm{\Delta }$, the relaxation rate exhibits a peak below the ferromagentic gap induced by emission of sound modes with velocity $v_{\mathrm{s}}$. Parameters used: $n{a}^2=0.03, T/J=0.2, \mathrm{\Delta }/J=0.1, a=0.3\mathrm{nm}$, and $d=20\mathrm{nm}$.

Figure 3

(a) ${\gamma }_{\mathit{k}}\left(z\right)$ plotted for different values $\left|\mathit{k}\right|/m{v}_{\mathrm{th}}=0,1,5$ (increasing darkness). Indicated with a dashed line is the linear ${\gamma }_{\mathit{k}}\left(z\right)=z/8\pi$ obtained from the classical Boltzmann equation. (b) ${\gamma }_{\mathit{k}}\left(z\right)$ exhibits a weak dependence on $k$, as shown for $z=1$. At most, ${\gamma }_{\mathit{k}}\left(z\right)$ varies by a factor of $\sim 2.5$ as $k$ is varied. In our calculations, we take the average of ${\gamma }_{\mathit{k}}$ over $\mathit{k}$ space.

Figure 4

(a) In addition to the sound mode, off-resonant processes can also give a finite contribution to ${\chi }_{-+}$ below the magnon gap. (b) Sunrise diagram contributing to the magnon self-energy of ${\chi }_{-+}$.