Quantum damping of skyrmion crystal eigenmodes due to spontaneous quasiparticle decay

Skyrmion crystals support chiral magnonic edge states akin to electronic quantum Hall edge states. However, magnonic topology relies on the harmonic approximation, neglecting ubiquitous magnon-magnon interactions that yield a finite zero-temperature quantum damping. We demonstrate that spontaneous quasiparticle decay in two-dimensional ferromagnetic skyrmion crystals is a delicate issue, with the quantum damping ranging over several orders of magnitude. Flat magnon bands cause exceptionally strong spontaneous decay at twice their energy. The resulting externally controllable energy-selective magnon breakdown is measurable not only by scattering but also by magnetic resonance experiments, probing the magnetically active anticlockwise, breathing, and clockwise modes. They exhibit distinct decay behavior, with the clockwise (anticlockwise) mode being the least (most) stable mode out of the three. The quantum damping of the topologically nontrivial anticlockwise mode is negligible, establishing the harmonic theory as a trustworthy approximation at low energies, implying excellent prospects of topological magnonics in skyrmion crystals.

Figure 1

Sketch of spontaneous magnon decay in a Néel skyrmion crystal sheet. Arrows in the background indicate the magnetic texture of the SkX, with color indicating the in-plane component of the texture. White spheres and wavy lines indicate propagating magnons. A magnon incoming from the left spontaneously decays into two other magnons. This number nonconserving process is brought about by the noncollinear texture and gives rise to a zero-temperature quantum damping of the SkX's eigenmodes.

Figure 2

Low-energy portion of the $M=8$ SkX magnon spectrum dependent on magnetic field $b$. (a)–(c) Single-particle magnon energies ${\varepsilon }_{\mathit{k},\nu }$, whose blue/gray/red color encodes negative/zero/positive magnetic moment ${\mu }_{\mathit{k},\nu }$. (d)–(f) Two-magnon DOS ${\mathcal{D}}_{\mathit{k}}^{2\mathrm{m}}\left(\varepsilon \right)$ (blue, zero; yellow, maximal), with red arrows indicating selected points where ${\varepsilon }_{\mathit{k},\nu }$ (white lines) crosses regions of large ${\mathcal{D}}_{\mathit{k}}^{2\mathrm{m}}\left(\varepsilon \right)$. (g)–(i) Spectral function $A_{\mathit{k}}\left(\varepsilon \right)$ for $S=1$, with sharp yellow/broad blue quasiparticle peaks indicating undamped/damped magnons. Strong damping is found for the crossing points (red arrows).

Figure 3

Magnon decay at the Brillouin zone center ($\mathit{k}=\mathit{0}$) of an $M=8$ SkX dependent on magnetic field $b$. (a) Two-magnon DOS ${\mathcal{D}}_{\mathit{0}}^{2\mathrm{m}}\left(\varepsilon \right)$ (blue, zero; yellow, maximal), with red arrows indicating selected points where the eigenmode energies ${\varepsilon }_{\mathit{0},\nu }$ (white lines) cross regions of large ${\mathcal{D}}_{\mathit{0}}^{2\mathrm{m}}\left(\varepsilon \right)$. (b) Spectral function $A_{\mathit{0}}\left(\varepsilon \right)$ for $S=1$, with sharp yellow/broad blue quasiparticle peaks indicating undamped/damped magnons. Strong damping is found for the crossing points (red arrows). (c) Spontaneous damping ${\mathrm{\Gamma }}_{\mathit{0},\nu }^{\mathrm{spon}}$ of the three magnetically active modes reveals that the clockwise (anticlockwise) mode is the least (most) stable.

Figure 4

Decay channel analysis of an $M=8$ SkX at $b/JS=0.5$ and $\mathit{k}=\mathit{0}$. Histograms show the decay amplitude $C^{\lambda \mu \leftarrow \nu }$ for $\lambda ,\mu =1,...,10$ and (a) $\nu =3$, anticlockwise mode (A); (b) $\nu =5$, breathing mode (B); and (c) $\nu =6$, clockwise mode (C). The histograms are symmetric due to decay product symmetry ($\lambda \leftrightarrow \mu$; cf. Appendix pp1). Red squares indicate the subsection of decay channels that could, in principle, fulfill energy conservation. By comparing the amplitudes of these channels between the three modes, the clockwise (anticlockwise) mode is found to interact strongly (weakly) with other modes. This explains the two order of magnitude difference in the damping [cf. Fig. 3].

Figure 5

Triangular lattice with indicated nearest-neighbor bond vectors ${\mathit{\delta }}_i$ and respective DM vectors ${\mathit{D}}_{{\mathit{\delta }}_i}$ ($i=1,...,6)$. The lattice constant is $a$.

Figure 6

(a) Spin spiral ordering vector $a{k}_x^{★}$ dependent on $D/J$ ($a$ lattice constant). (b) Relative error between the exact solution for $k_x^{★}$ and the approximation in Eq. (A10) dependent on $D/J$.

Figure 7

Feynman diagrams of order-$1/S$ self-energies. Gray (Black) lines denote external (internal) legs and green/red/blue circles denote interaction vertices. The green circle is the two-in–two-out four-magnon vertex (not given) and the red/blue circles are the decay ($\mathcal{V}$) and source ($\mathcal{W}$) three-magnon vertices (or their conjugates) given in Eqs. (A44) and (A46), respectively. (a) First-order Hartree-Fock corrections due to ${\widehat{H}}_4$ that yield a static real self-energy. (b) Second-order tadpole diagrams due to ${\widehat{H}}_3$ that also yield a static real self-energy. (c) Second-order bubble diagrams due to ${\widehat{H}}_3$ that yield a dynamic self-energy. However, only the first two diagrams contribute to the imaginary part of the self-energy.

Figure 8

Same as Fig. 2 but for additional magnetic field values.

Figure 9

Decay channel analysis of an $M=8$ SkX at $b/JS=0.5$ and $\mathit{k}=\mathit{0}$. (a)–(j) Decay channel efficiency $C^{\lambda \mu \leftarrow \nu }$ for $\lambda ,\mu ,\nu =1,...,10$: (a) $\nu =1$, gyrotropic mode, $l=1$; (b) $\nu =2$, elliptic mode, $l=2$; (c) $\nu =3$, anticlockwise mode; (d) $\nu =4$, triangular mode, $l=3$; (e) $\nu =5$, breathing mode; (f) $\nu =6$, clockwise mode; (g) $\nu =7$, quadrupolar mode, $l=4$; (h) $\nu =8$; (i) $\nu =9$; and (j) $\nu =10$, pentagonal mode, $l=5$. Small blue/large red squares indicate small/large $C^{\lambda \mu \leftarrow \nu }$. The symbol $\times$ indicates exactly zero $C^{\lambda \mu \leftarrow \nu }$. Notice the decay product symmetry $\lambda \leftrightarrow \mu$.

Figure 10

Decay channel efficiencies $C^{\lambda \mu \leftarrow \nu }$ for the (a) C mode, (b) B mode, and (c) A mode dependent on magnetic field $b$. For each of the three modes, we used relative color scales, normalized to the largest $C^{\lambda \mu \leftarrow \nu }$ (large red squares).

Figure 11

Damping of ABC modes of an $M=8$ SkX at $b/JS=0.485$ dependent on temperature $T$. (a) Total damping ${\mathrm{\Gamma }}_{\mathit{0}}^{\mathrm{total}}\left(T\right)={\mathrm{\Gamma }}_{\mathit{0}}^{\mathrm{spon}}+{\mathrm{\Gamma }}_{\mathit{0}}^{\mathrm{decay}}\left(T\right)+{\mathrm{\Gamma }}_{\mathit{0}}^{\mathrm{coll}}\left(T\right)$, (b) damping ${\mathrm{\Gamma }}_{\mathit{0}}^{\mathrm{decay}}\left(T\right)$ due to thermally activated decays, and (c) damping ${\mathrm{\Gamma }}_{\mathit{0}}^{\mathrm{coll}}\left(T\right)$ due to thermally activated collisions. Also shown is the magnon band structure in the vicinity of the Brillouin zone center with (d) the exemplary decay process and (e) the exemplary collision process.

Figure 12

Magnetic moment of a SkX's translational Goldstone mode (a) along high-symmetry paths and (b) dependent on the distance from the Brillouin zone center.

Figure 13

Ratio of quantum corrections to the magnetization $\delta M_0$ and the classical ground-state magnetization $M_0$ dependent on $D/J$ (or linear skyrmion size $M$). We set $b/JS=0.5{(D/J)}^2$.

Figure 14

Integrated three-magnon vertex dependent on $D/J$ (bottom abscissa) and linear skyrmion size $M$ (top abscissa). We set $b/JS=0.5{(D/J)}^2$.