Spin demons in d-wave altermagnets

TL;DR

This work identifies a spin demon in $d$-wave altermagnets, a novel out-of-phase collective mode of the two spin densities that can be underdamped because it sits outside the particle-hole continuum of one spin species. Using random-phase approximation for spin-resolved response functions with an anisotropic spin-split Fermi surface, the authors show the spin demon appears as a zero of the longitudinal dielectric function $\epsilon(q,\omega)$ outside the spin-down continuum, yielding a sharp peak in ${\rm Im}\,\chi_{S_zS_z}$. The mode propagates with a velocity $\omega_d(\mathbf q)=v_d\,\eta_{\min}(\theta)\,q$ in the altermagnetic spin-split plane, and its damping $\gamma$ determines a high quality factor $Q=\omega_d/\gamma$ that remains large for realistic parameters; the demon carries a magnetic moment $\mu_d$ that changes sign with propagation direction, reflecting the $d$-wave symmetry. The spin demon persists in both 3D and 2D altermagnets, with analytical expressions in 2D showing enhanced robustness, and is predicted to be detectable via spin-sensitive probes like SPEELS or polarized Raman, enabling direct experimental access to this new spin-plasmonic excitation.

Abstract

Demons are a type of plasmons, which consist of out-of-phase oscillations of electrons in different bands. Here, we show that $d$-wave altermagnets, a recently discovered class of collinear magnetism, naturally realize a spin demon, which consists of out-of-phase movement of the two spin species. The spin demon lives outside of the particle-hole continuum of one of the spin species, and is therefore significantly underdamped, reaching quality factors of $>10$. We show that the spin demon carries a magnetic moment, which inherits the $d$-wave symmetry. Finally, we consider both three and two dimensional $d$-wave altermagnets, and show that spin demons exists in both.

Figures (5)

• Figure 1: The imaginary part of the spin–spin response function, $\mathop{\mathrm{Im}}\nolimits[\chi_{S_zS_z}(\bm q^*,\omega)]$, where $\bm q^*=q\ (\cos\theta,\sin\theta,0)$, rotated in the altermagnetic spin-split plane with a fixed $q=0.05k_{F}$. The angular direction encodes $\theta$ and the radial axis the frequency $\omega$. The colors on the ring indicate the projected spin species, with red (blue) spin up (down). The spin demon is the sharp resonance that follows the four-fold rotational symmetry of the $d$-wave altermagnet. The anisotropically spin-split Fermi surfaces are schematically shown at the origin (not to scale).
• Figure 2: (a) The real and imaginary part of the dielectric function, for $\bm q^* = q \hat{\bm x}$, with $q=0.05k_{F}$. The zeros of the real part correspond to resonances, the imaginary part determines their damping. The spin demon is the second zero. The blue and red shading indicate where the spin-down and spin-up particle-hole continua is non-zero. (b) The imaginary part of the spin–spin response function, $\mathop{\mathrm{Im}}\nolimits[\chi_{S_zS_z}(\bm q,\omega)]$, for $\bm q=q\hat{\bm{x}}$, showing the existence of a spin demon with a high quality factor. The vertical dotted line corresponds to the $q$ used in (a). In both (a,b), the dashed lines indicate the spin-resolved particle-hole continua edges, $\omega_{+\sigma}$.
• Figure 3: The spin demon velocity (top) and quality factor (bottom), as a function of angle $\theta$. The numerical solutions (solid), are obtained by numerically finding the zeros and corresponding derivatives from the full dielectric function; the analytical solutions (dashed) follow from numerically solving \ref{['eq:zeros-vs']}.
• Figure 4: Numerically obtained spin demon resonance frequency $\omega_d$ as a function of $\bm q^*=q\ (\cos\theta, \sin\theta,0)$, with $q=0.05k_{F}$, rotated in the altermagnetic spin splitting plane with angle $\theta$. The color corresponds to the magnetic moment, showing the $d$-wave character. Obtained by numerically finding the zeros and evaluating \ref{['eq:mud']} from the full dielectric function. Along $x$ and $y$, the magnetic moment is approximately $\pm0.025\mu_B$. The colors on the ring indicate the projected spin species.
• Figure 5: In two dimensions: (a) $\mathop{\mathrm{Im}}\nolimits[\chi_{S_zS_z}(\bm q^*,\omega)]$, The angle indicates $\theta$ and the radial axis the frequency $\omega$. (b) The spin demon velocity (top) and quality factor (bottom) as a function of rotation angle $\theta$.