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PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

Quantum damping of skyrmion crystal eigenmodes due to spontaneous decay

Alexander Mook , Jelena Klinovaja , and Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland

(Received 2 March 2020; revised 4 June 2020; accepted 29 July 2020; published 25 September 2020)

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 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.

DOI: 10.1103/PhysRevResearch.2.033491

I. INTRODUCTION Herein we explore the many-body quantum physics of in two-dimensional ferromagnetic SkXs and estab- crystals (SkXs), as depicted in Fig. 1, lish them as materials of extremes. Some magnon bands are are attracting much attention. These arrays of topologically flat akin to Landau levels, giving rise to sharp peaks in the nontrivial magnetic whirls appear in bulk, multilayers, and density of states (DOS) of the two-magnon continuum, into thin films, at elevated and zero temperature, in metals and which single magnons decay. Manipulating these peaks by insulators [1–18]. Besides numerous fundamental discover- a magnetic field allows for a field-tunable energy-selective ies [19–28], they have spawned visions of future spintronic magnon breakdown detectable by scattering or absorption [29–31], logic [32–37], quantum information [38–40], and experiments. We reveal that the anticlockwise (clockwise) magnonic applications [41–53]. The three magnetically active mode is the most (least) stable out of the three magnetically SkX eigenmodes, associated with an anticlockwise, breath- active modes. Even for ultrasmall skyrmions, the damping of ing, or clockwise motion of the skyrmion core [23], belong the topologically nontrivial anticlockwise mode is negligible, to the three lowest topologically nontrivial magnon bands establishing excellent prospects of low-energy topological [44,48,52], supporting chiral edge magnons [54–56], akin magnonics in SkXs. to electronic quantum Hall edge states. Since these chiral edge magnons are innately free of Joule heating, immune to backscattering, and robust against considerable disor- der, they are proposed as “information highways” in the field of topological magnonics [57]. However, such fun- damental physics reveals itself only at low temperatures, necessitating pushing the frontiers of skyrmionics to the ul- traquantum regime, a trend that began only recently [58–72]. Importantly, even at zero temperature, interactions between magnons are not frozen out and spontaneous quasiparti- cle decay (SQD) [73–110], as sketched in Fig. 1, dresses magnons with intrinsic lifetimes, potentially spoiling the con- cept of topological magnonics, which relies on the harmonic approximation. FIG. 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 tex- ture. White spheres and wavy lines indicate propagating magnons. A Published by the American Physical Society under the terms of the magnon incoming from the left spontaneously decays into two other Creative Commons Attribution 4.0 International license. Further magnons. This number nonconserving process is brought about by distribution of this work must maintain attribution to the author(s) the noncollinear texture and gives rise to a zero-temperature quantum and the published article’s title, journal citation, and DOI. damping of the SkX’s eigenmodes.

2643-1564/2020/2(3)/033491(19) 033491-1 Published by the American Physical Society MOOK, KLINOVAJA, AND LOSS PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

II. THEORY ν = 1,...,N is the band index and k the crystal (see Appendix A2). The cubic decay processes, within which Interacting spins Sˆ , localized at lattice sites with posi- r a magnon in state (p,ν) decays into two magnons in states tion vector r, are effectively described by Hamiltonians ,λ ,μ 1 z (k ) and (q ), respectively, read (see Appendix A2) ˆ = ˆ · , · ˆ + ˆ H 2 r,r Sr Ir r Sr b r Sr , with interaction matri- √ ces Ir,r and a Zeeman energy due to magnetic field b. p=k+q The classical ground state of Hˆ , obtained by treating the d S 1 λμ←ν † † Hˆ = √ V , ← bˆ ,λbˆ ,μbˆ p,ν + H.c. ˆ 3 3 N 2!1! k q p k q Sr’s as classical vectors Sr in R , defines local reference u λ,μ,ν k,q,p { x, y, z} z frames er er er , with er along the classical ground-state (1) direction. Excitations above this ground state are captured λμ←ν by the Holstein-Primakoff√ transformation [111] to (N the number of unit cells). The vertex V comprises (†) ˆ = ˆ − + † ˆ + + − † z ± = u k,q←p aˆr : Sr √S( fraˆrer aˆr frer ) (S aˆr aˆr)er, with er the interaction strength and the Hermitian conjugate part a x ± y / ˆ = − (er ier ) 2 and spin length S. Expanding fr (1 magnon coalescence. † / 1/2 / ˆ = ∞ ˆ aˆr aˆr 2S) in powers of 1 S leads to H t=0 Ht , where To account for Hˆ , we perform second-order many-body ˆ ∝ 2−t/2 3 Ht O(S ) includes bosonic operators to the tth power. perturbation theory. Concentrating on dynamical 1/S cor- ˆ Sub-Hamiltonian H0 gives the classical ground-state en- rections to the spectrum, we approximate the single- ˆ ˆ −1 ergy, H1 vanishes if the magnetic texture is stable, H2 Green’s function by G (ε) ≈ ε − ε ,ν + i ,ν , from which ˆ k,ν k k describes free magnons, and Ht>2 denotes interactions. In par- we obtain the spectral function A ,ν (ε) ≈−ImG ,ν (ε)/π of ˆ k k ticular, H3 comprises number nonconserving three-magnon band ν. Evaluating single-bubble diagrams within the on-shell interactions. approximation, we derive the spontaneous zero-temperature ˆ Often, however, linear spin-wave theory (H2) is sufficient damping (see Appendix A3) because the single-particle sector is disconnected from many- π particle sectors in one of the following limits: (1) classical spon λμ←ν 2  = V δ(ε ,ν − ε ,λ − ε − ,μ). →∞ / k,ν 2N q,k−q←k k q k q spins S because interactions cause at least 1 S cor- u q λ,μ rections to the spectrum, (2) low temperatures T that freeze (2) out thermally activated interactions, or (3) large fields b that polarize the and energetically separate the m-magnon = , ,... manifolds, whose energies grow with mb (m 1 2 ). III. RESULTS Intriguingly, none of the above limits is applicable to SkXs. (1) Flat bands cause decay singularities at twice their energy, First, we review the results of harmonic theory [44,48,52]. only regularized at the same 1/S order that introduces damp- Figures 2(a)–2(c) show the dispersion of the lowest free- ing in the first place. Thus, there is no 1/S smallness in the magnon bands εk,ν of an M = 8 SkX dependent on b (D/J = damping and large remnants of quantum effects in effectively 1). We identify flat modes that derive from single-skyrmion classical spin systems are expected, an effect otherwise known modes associated with an lth-order polygon deformation in from kagome antiferromagnets [98,99]. (2) Three-magnon in- real space [117][l = 2, elliptic (E); l = 3, triangular (T); = = teractions (Hˆ3) cause spontaneous decay even at T = 0[95]. l 4, quadrupolar (Q); l 5, pentagonal mode (P); etc.]. (3) destroys the SkX and cannot be used. Moreover, there are dispersive modes, the lowest four of This prima facie case for studying magnon-magnon in- which coincide with the gyrotropic (G), anticlockwise (A), teractions in SkXs calls for a nonlinear spin-wave theory of breathing (B), and clockwise (C) modes, respectively, at the chiral . Here we consider a two-dimensional chiral Brillouin zone center [48,52] [cf. labels in Fig. 2(d)]. The G magnet on the triangular lattice (in the xy plane) with lattice mode derives from the translational l = 1 mode of a single constant a. Its interaction matrices with elements (Ir,r )mn = skyrmion [118]. The A, B, and C modes carry a nontrivial − δ +  p δ  Chern number and support chiral edge magnons [44,48,52]. J mn mnpDr,r ( mn and mnp are the Kronecker delta and the Levi-Civita symbol, respectively) include isotropic The color of the bands in Figs. 2(a)–2(c) indicates the symmetric ferromagnetic exchange J > 0 and antisymmet- magnonic out-of-plane magnetic moment μk,ν =−∂εk,ν /∂b. ric exchange denoted by a Dzyaloshinskii-Moriya (DM) A positive (negative) μk,ν means that an increasing field [112,113] vector Dr,r = Dez × er,r of length D that complies shifts the modes towards lower (higher) energies. Flat modes μ with interfacial inversion symmetry breaking [114,115]; ez is a carry large negative k,ν because of additional angular mo- unit vector in the z direction and er,r one in the bond direction. mentum associated with l. Hence, upon increasing the field, For D = 0√ and b = 0, the√ ground-state spin spiral has a they overtake the dispersive modes, which strongly influences λ ≈ π −1 / spon pitch of 3 aarctan ( 3D 2J) (see Appendix A1). SQD. To see so, note that k,ν is a weighted two-magnon > ≈ . 2/ 2m 1 Once b bc,1 0 2D J [116], a Néel SkX forms. A com- DOS D (ε) = λ,μ δ(ε − εq,λ − εk−q,μ)atε = εk,ν . λ = = k Nu q mensurate SkX with Ma (M an integer) and N In Figs. 2(d)–2(f) the εk,ν ’s are overlaid with colormaps that 2 / ≈ D2m ε D2m ε M spins per magnetic unit cell is obtained for D J show k ( ). Besides extended regions of moderate k ( ), tan(2π/M). Field polarization is reached at b > bc,2 ≈ several horizontal features, i.e., sharp yellow lines, at twice 0.8D2/J [116]. the energy (or at the sum of two energies) of flat modes are Numerically, we proceed as follows. We choose M, com- identified. If a single-magnon branch crosses a region of large D2m ε pute D and J, and equilibrate a random spin configuration k ( ), it is “in resonance” with flat modes and the kinemat- at T = 0. We obtain the single-particle energies εk,ν by di- ically allowed phase space for SQD, i.e., the number of decay ˆ ˆ(†) agonalizing H2 in terms of normal mode bosons bk,ν , where channels that fulfill both energy and momentum conservation,

033491-2 QUANTUM DAMPING OF SKYRMION CRYSTAL … PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

Magnetic field

b /(JS)=0.3 b /(JS)=0.5 b /(JS)=0.7 4 (a) (b) l = 5(P) (c)

3

) l = 4(Q) JS ( c 2 lockwise breathing l = 3(T) Energy Γ K anticlockwise ticlockwis 1 an e M l = 2(E)

gyrotropic 0 4 (d) (e) (f)

3 anticlockwise ) JS ( 2 Energy 1 FIG. 3. Magnon decay at the Brillouin zone center (k = 0)of an M = 8 SkX dependent on magnetic field b. (a) Two-magnon 0 DOS D2m (ε) (blue, zero; yellow, maximal), with red arrows in- 4 0 (g) (h) (i) dicating selected points where the eigenmode energies ε0,ν (white D2m ε ε lines) cross regions of large 0 ( ). (b) Spectral function A0( )for 3 S = 1, with sharp yellow/broad blue quasiparticle peaks indicating undamped/damped magnons. Strong damping is found for the cross- J spon ing points (red arrows). (c) Spontaneous damping 0,ν of the three 2 magnetically active modes reveals that the clockwise (anticlockwise)

Energy mode is the least (most) stable.

1

0 MΓ KMMΓ KMMΓ KM in Figs. 2(b), 2(e), and 2(h) (cf. the upper right red arrow). However, a slight magnetic-field detuning of the resonance FIG. 2. Low-energy portion of the M = 8 SkX magnon spectrum condition leads to a reappearing quasiparticle peak. To con- dependent on magnetic field b. (a)–(c) Single-particle magnon ener- firm so, see Appendix B1. gies εk,ν , whose blue/gray/red color encodes negative/zero/positive Besides flat-mode resonances, one identifies several re- μ D2m ε D2m ε magnetic moment k,ν . (d)–(f) Two-magnon DOS k ( ) (blue, gions with large damping but moderate k ( ), associated zero; yellow, maximal), with red arrows indicating selected points with decays into (at least) one dispersive mode. As an exam- ε D2m ε where k,ν (white lines) crosses regions of large k ( ). (g)–(i) ple, consider the C mode in Fig. 2(h), indicated by the green Spectral function Ak (ε)forS = 1, with sharp yellow/broad blue arrow. Its strong damping is not related to a particularly large quasiparticle peaks indicating undamped/damped magnons. Strong D2m ε k ( )inFig.2(e). This mechanism is the prime cause of damping is found for the crossing points (red arrows). high-energy magnon damping in Figs. 2(g)–2(i), reflecting the abundance of kinematically allowed decay channels. Since all decay channels that contain at least one flat is particularly large. Selected instances of such crossings are mode as decay product exhibit a strong field dependence, the marked by red arrows and pronounced spon is expected. k,ν SQD is field tunable and highly energy selective, which sets We confirm this prediction by calculating spon and k,ν SkXs apart from other SQD platforms. This finding can be ε = N ε Ak( ) ν=1 Ak,ν ( ), the latter of which is shown in confirmed by inelastic or resonant x-ray-scattering Figs. 2(g)–2(i). Stable magnons appear as sharp yellow quasi- experiments, which measure the dynamical structure factor particle peaks, while strongly damped magnons experience [119]. A rich magnetic-field dependence of linewidths is ex- considerable lifetime broadening (broad blue features). For pected. To pursue this thought further, we continue with an example, consider the regions of pronounced damping marked analysis of uniform excitations (k = 0), which can even be by red arrows in Fig. 2(g), which can be traced back to studied by absorption experiments [120–123]. For the mag- D2m ε k ( )inFig.2(d). These flat-band resonances can be so netic field normal to the SkX, only the ABC modes are active large that the spectral weight is almost completely wiped out, [23]. However, inclined fields admit additional active modes as exemplified by the pentagonal l = 5 mode at b/JS = 0.5 [124].

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(a) Anticlockwise mode

(b) Breathing mode

P (c) Clockwise mode 9 8 Q C P P B 8 9 9 λ T Q 8 A B C Q E T μ E A C 9 P G G P λ B 8 9 T C Q 8 A B λμ←ν Q E A T μ Decay amplitude C (meV2) C P G E B 8 9 G λ T Q A B C E A T μ 0.0 0.005 0.01 G G E

FIG. 4. Decay channel analysis of an M = 8SkXatb/JS = 0.5andk = 0. Histograms show the decay amplitude Cλμ←ν for λ, μ = 1,...,10 and (a) ν = 3, anticlockwise mode (A); (b) ν = 5, breathing mode (B); and (c) ν = 6, clockwise mode (C). The histograms are symmetric due to decay product symmetry (λ ↔ μ; cf. Appendix A2). 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(c)].

D2m ε Figure 3(a) shows 0 ( ) dependent on b. Again, strong The results discussed so far apply to zero temperature. In damping is found when ε0,ν crosses a region of large Appendix B4 we extend our analysis to finite temperatures: D2m ε k T/J = . b/JS = . k 0 ( ). Several such encounters are highlighted by red ar- At B 0 2 and 0 485 ( B the Boltzmann con- rows, from which a rich structure of strong quasiparticle stant) the thermal damping of the C mode is still only half peak broadening is expected. Indeed, A0(ε)inFig.3(b) re- of its quantum damping. Thus, even the quantum dynamics veals that high-energy magnons exhibit field intervals within of SkXs stabilized by moderate thermal fluctuations may be which their quasiparticle peak disappears and reappears. In dominated by SQD. However, for consistency of spin-wave contrast, low-energy magnons exhibit negligible quantum theory, kBT JS must hold (see Appendix B5). damping. There are considerable differences in damping between the IV. DISCUSSION AND CONCLUSION ABC modes [cf. Fig. 3(c)]. The SQD is particularly strong for the C mode, whose damping (0.2J) is on par with frustrated Spontaneously decaying magnons cause field-tunable in- quantum antiferromagnets [84,88,95]. In contrast, the damp- trinsic zero-temperature damping of SkX eigenmodes. For ing of the B (A) mode is a factor of 10 (100) smaller. These applications in topological magnonics, this damping must D2m ε differences cannot solely be explained in terms of 0 ( )but not be larger than the average band gap ≈ 12SJ/N ≈ Vλμ←ν . / 2 must be looked for in k,q←p. 0 3[JS(D J) ], hosting chiral edge states. With the topolog- spon 4 We analyze each decay channel ν → (λ, μ) by measuring ically nontrivial ABC modes’ damping  ≈ FX J(D/J) , − X − λμ←ν 1 λμ←ν 2 with X ∈{A, B, C} and F = 10 3, F = 10 2, and F = its integrated decay amplitude C ≡ |V ,− ← | . A B C Nu q q q 0 −1 spon/ ≈ It encodes dynamical rather than kinematic details. Results 10 (see Appendix B6), the crucial ratio reads X / . / 2 / for 1000 decay channels, involving the ten lowest modes at FX (0 3S)(D J) 1 for typical values D J 1. For b/JS = 0.5, are presented in Appendix B2. Here we focus on thin films of the skyrmion-hosting compound Cu2OSeO3, spon/ ≈ . / . μ = . × −3 the decay of ABC modes (Fig. 4). we estimate C 5 6 neV 4 7 eV 1 2 10 us- = / ≈ / ≈ . The A mode [ν = 3, Fig. 4(a)] exhibits a small interac- ing S 1, J kB 50 K, and D J 0 06 [8]. Even for small tion with other modes, especially those that could possibly skyrmions (D/J = 1), quantum damping of the A and B fulfill energy conservation (G and E). Thus, the A mode is modes is negligible. Hence, SkXs are an excellent platform the most stable out of the ABC modes for both kinematic for low-energy topological magnonics, which is at variance and dynamic reasons. The B mode [ν = 5, Fig. 4(b)] prefers with other topological magnets [125–127] whose chiral edge channel B → (B, B), which, however, can never obey energy magnons appear at high energies of the order of the exchange conservation, and channels involving the G and/or A mode, energy. causing the moderate damping found in Fig. 3. In contrast, At higher energies, SkXs exhibit considerable SQD, which α ≈ channels with polygon modes E, T, and Q as decay products yields a quantum correction to the classical damping k,ν are suppressed by two orders of magnitude. Finally, being αεk,ν ∝ O(S) due to phenomenological Gilbert damping α. highest in energy, the C mode [ν = 6, Fig. 4(c)] has several Skyrmion crystals are unique in that not only neutron scatter- kinematic possibilities to decay. On top of that are pronounced ing but also magnetic resonance experiments can address the instabilities towards the A and B modes. Thus, the C mode lifetime broadening due to SQD. For the magnetically active 2 is the least stable among the ABC modes. These findings C mode, with energy εC ∼ JS(D/J) , the relative importance ≡ spon/α ≈ are corroborated by the nontrivial b dependence of the ABC of quantum to classical damping reads RC C C modes’ interaction strengths (see Appendix B3): While that 10−1(αS)−1(D/J)2. Since ultralow α ∼ 10−4–10−3 is possi- of the C mode (and B mode) increases as the field increases, ble in both metals [128–137] and insulators [138], quantum that of the A mode decreases. damping may dominate over classical damping (RC > 1), e.g.,

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−4 RC ≈ 3.6inCu2OSeO3 with α = 10 [138], a prediction that may be experimentally tested in the recently discovered low-temperature SkX [139]. Our strictly two-dimensional considerations apply both to thin films (cf. Appendix B7) and quasi-two-dimensional lay- ered structures, e.g., van der Waals magnets [140–142] and Janus layers [143,144]. Although we concentrated on Néel skyrmions stabilized by interfacial DM interaction, our results can be carried over one to one to Bloch skyrmions due to bulk DM interaction because neither the harmonic magnon bands nor their damping is influenced by the skyrmion helicity. Additional anisotropies, exchange frustration [49,145–151], and dipolar and four-spin interactions [152] are expected to FIG. 5. Triangular lattice with indicated nearest-neighbor bond vectors δ and respective DM vectors Dδ (i = 1,...,6). The lattice merely quantitatively renormalize the quantum damping. i i constant is a.

ACKNOWLEDGMENTS energy can be rewritten as We thank Sebastián Díaz for helpful discussions. This work 6 − ·δ was supported by the Georg H. Endress Foundation, the Swiss = ik i , Ik e Iδi (A3) National Science Foundation, and NCCR QSIT. This project i=1 received funding from the European Union’s Horizon 2020 where the δ ’s are the vectors to the six nearest neighbors on research and innovation program (ERC Starting Grant Agree- i the triangular lattice, as shown in Fig. 5; explicitly they read ment No. 757725). − 1 − 1 1 2 2 δ1 = a , δ2 = a √ , δ3 = a √ , (A4) APPENDIX A: DETAILS OF THEORETICAL 0 3 − 3 DERIVATIONS 2 2 δ4 =−δ1, δ5 =−δ2, and δ6 =−δ3. (We denoted the lattice 1. Classical magnetic ground state constant by a.) The respective interaction matrix ⎛ ⎞ Before studying the dynamics of a spin Hamiltonian Hˆ ,we y −J 0 −Dδ need to find the classical magnetic ground state, assuming that i ⎜ x ⎟ ˆ 3 δ = ⎝ 0 −JD⎠ spin operators Sr are replaced by spin vectors Sr in R .For I i δi (A5) J > 0, neighboring spins prefer ferromagnetic alignment that Dy −Dx −J δi δi competes with the spiralization tendency due to DM interac- tion. In the absence of a magnetic field, the classical magnetic contains the DM vectors √ √ ground state is thus a spin spiral. To determine its pitch we 0 − 3 3 2 perform a spatial Fourier transformation from coordinates r to Dδ = D , Dδ = D , Dδ = D 2 , (A6) 1 1 2 − 1 3 − 1 (crystal) momenta k, 2 2 =− =− =− Dδ4 Dδ1 , Dδ5 Dδ2 , and Dδ6 Dδ3 . They are indi- 1 ik·r Sr = √ e Sk, (A1) cated by red arrows in Fig. 5. N s k To find the energy-minimizing spin configuration, we de- termine the eigenvalues of where Ns is the total number of spins. Thus, the classical ⎛ ⎞ ground-state energy in zero field is recast as −Zk 0 −Yk ⎜ ⎟ = ⎝ − ⎠, 1 Ik 0 Zk Xk (A7) = · · H Sr Ir,r Sr (A2a) − − 2 Yk Xk Zk r,r which read 1 i(k+k )·r ik·Δr  = Sk · (e e Ir,r+Δr) · S (A2b) k V , =−Z , V±, =−Z ± |X |2 +|Y |2, (A8) 2Ns 0 k k k k k k k,k r,Δr with 1 −ik·Δr = Sk · e IΔr ·S−k. (A2c) 3 2 = · δ , k Δr Zk 2J cos(k i ) (A9a) i=1 Ik 3 + · x i(k k ) r = δ = + Δ =− · δ , We use r e Ns k,−k and r r r, introduc- Xk 2i Dδ sin(k i ) (A9b) i ing the distance vector Δr = r − r.WesetIr,r = Ir,r+Δr = i=1 IΔr, reflecting spatial homogeneity of magnetic interactions. 3 Since magnetic interactions take place only between near- Y =−2i Dy sin(k · δ ). (A9c) k δi i est neighbors, the Fourier kernel of the classical ground-state i=1

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former update each spin according to B · Sold Snew = 2 r r B − Sold, (A13) r 2 r r |Br|

with Br =−∂H/∂Sr the effective magnetic field at site r.For the latter, we integrate the LLG equation without a precession term, i.e., α S˙ =− S × (S × B ), (A14) r 1 + α2 r r r

/ utilizing the Heun integration scheme [153]. We work in the FIG. 6. (a) Spin spiral ordering vector akx dependent on D J (a overdamped limit with a Gilbert damping α = 0.99. lattice constant). (b) Relative error between the exact solution for kx and the approximation in Eq. (A10) dependent on D/J. 2. Spin-to- transformation Once the magnetic ground state with N spins per unit The ordering vector k of the ground-state texture is given x y z cell is found, we assign a coordinate system {eμ, eμ, eμ} to by the minima of the lowest eigenvalue V−, . Due to the k each basis site (μ = 1,...,N). The local reference frames sixfold rotational symmetry of the triangular lattice, there are z = , are chosen such that eμ points along the local ground-state six minima; one solution reads k (kx 0), with the others direction of spin S + .Herer is the position vector to the to be obtained by rotation. The numerically obtained solution r rμ / magnetic unit cell and rμ is the position vector to the μth basis for kx , shown in Fig. 6(a) dependent on D J, is very well approximated by site within the unit cell. Then the spin Hamiltonian may be √ recast as 2 3 D 1 ≈ √ . ˆ = ˆ · · ˆ + · ˆ . kx arctan (A10) H Sr+rμ Ir+rμ,r +rν Sr +rν b Sr+rμ 3a 2 J 2 r,r rμ,rν r rμ Figure 6(b) shows the relative error between Eq. (A10) and the (A15) exact solution. Even for ratios as large as D/J = 1, the relative . For the sake of generality, we allow for a magnetic field b in errorissmallerthan02%.FromEq.(A10), we obtain a spin an arbitrary direction. Here Ir+rμ,r +rν is the interaction matrix spiral pitch of between the μth and νth spins belonging to the magnetic unit  √ − cells at r and r , respectively. π √ 1 λ = 2 ≈ π 3 D . We proceed with a Holstein-Primakoff (HP) transforma- 3 a arctan (A11) kx 2 J tion √   Once the magnetic field is larger than a critical value, the ˆ = ˆ − + † ˆ + Sr+rμ S fr+rμ aˆr+rμ eμ aˆr+rμ fr+rμ eμ skyrmion crystal is approximately formed as a superposition   † z + − + , of a ferromagnetic component and three fundamental spin S aˆr+rμ aˆr rμ eμ (A16) spirals (neglecting higher harmonics), each pair of which √ ± = x ± y / encloses an angle of ±2π/3. Restricting our focus on com- where eμ (eμ ieμ) 2 and S is the spin length. The λ = bosonic operators obey the usual commutation relation mensurate textures, we require Ma to be an integer † [ˆa + , aˆ ] = δ , δ , . A Taylor expansion of multiple of a; thus M ∈ N. It follows that if a skyrmion crystal r rμ r +rν r r rμ rν with N = M2 spins per magnetic unit cell (i.e., per skyrmion) † 1/2 † + + is desired, the magnetic interaction parameters must obey aˆr+rμ aˆr rμ 1 aˆr+rμ aˆr rμ fˆ+ = 1 − = 1 − −··· √ r rμ 2S 2 2S D 2 3π 2π ≈ √ tan ≈ tan . (A12) (A17) J 3 M M leads to Hˆ = Hˆ0 + Hˆ1 + Hˆ2 + Hˆ3 + Hˆ4 +···, with sub- We note in passing that we use a triangular lattice be- scripts denoting the number of bosonic operators. Mathemat- cause it naturally lends itself to a description of a hexagonal ically,√ the HP transformation is an expansion in the order of skyrmion crystal. Nonetheless, similar results are expected for 1/ S, which can be made explicit by writing other two-dimensional lattices, e.g., the square lattice. The details of the structural lattice are irrelevant as long as a single √ 1 1 Hˆ = S SHˆ + SHˆ + Hˆ + √ Hˆ + Hˆ +··· . skyrmion’s diameter is large compared to the structural lattice 0 1 2 S 3 S 4 constant. (A18) Now that we approximately know the size of the skyrmion unit cell (M × M spins) for a given D/J ratio, the actual The primed Hˆ ’s are independent of S. Thus, when we later skyrmion texture within the unit cell is determined numeri- speak of “1/S corrections to the spectrum,” we mean correc- cally in a finite magnetic field. Starting from a random spin tions that are 1/S relative to Hˆ2. Those can come both from configuration, we alternate between microcanonical overre- Hˆ4 (within first-order perturbation theory) and Hˆ3 (within laxation steps and Landau-Lifshitz-Gilbert (LLG) steps. The second-order perturbation theory).

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a. Harmonic theory where the sum runs over all difference vectors Δr between After plugging Eqs. (A17) and (A16) into Eq. (A15), the magnetic unit cells. ε harmonic piece of Hˆ is found to read The free magnon energies k,ν are obtained from a parau-  nitary Bogoliubov diagonalization from HP bosons to normal S −− † +− modes ˆ = + + I + + I H2 aˆr rμ aˆr rν r+rμ,r +rν aˆr+rμ aˆr rν r+rμ,r +rν 2 r,r rμ,rν ˆ = ˆ = ˆ ,...,ˆ , ˆ† ,...,ˆ† T. Bk PkAk (bk,1 bk,N b−k,1 b−k,N ) (A26) † −+ † † ++ + aˆ + aˆ I + aˆ + aˆ I r rμ r +rν r+rμ,r +rν r rμ r +rν r+rμ,r +rν Thus, we can rewrite Eq. (A21) in the Bogoliubov eigenbasis  † zz † zz as − aˆ + aˆ + I − aˆ aˆ + I r rμ r rμ r+rμ,r +rν r +rν r rν r+rμ,r +rν 1 † − − † z ˆ = ˆ · † † 1 1 · ˆ , − aˆ aˆ + b · e , (A19) H2 Ak Pk (Pk ) HkPk Pk Ak (A27) r+rμ r rμ μ 2 k rμ † ˆ ˆ Ek Bk Bk ξζ ξ ζ I ≡ · + , + · ξ,ζ ∈{+, −, } where r+rμ,r +rν eμ Ir rμ r rν eν , with z . with the diagonal energy matrix We then perform a Fourier transformation ≡ † −1 −1 Ek (Pk ) HkPk Nu 1 = ε ,...,ε ,ε ,...,ε . = √ ik·(r+rμ ) , diag( k,1 k,N −k,1 −k,N ) (A28) aˆr+rμ e aˆk,μ (A20a) N u k The paraunitarity of Pk is expressed by N u † † 1 −ik·(r+rμ ) † = = , − . = √ , U Pk UPk diag(1N 1N ) (A29) aˆr+rμ e aˆk,μ (A20b) Nu k Here U is the bosonic metric, necessary to preserve bosonic with Nu the number of magnetic unit cells. In compact nota- commutation rules. In addition, 1N is the N × N unit matrix. tion, the harmonic Hamiltonian becomes Note the relation 1 † −1 = † . Hˆ = Aˆ · H · Aˆ , (A21) Pk UPk U (A30) 2 2 k k k k The above diagonalization procedure follows Ref. [154]. with Below we need the expressions † † T Aˆ k = (ˆak,1,...,aˆk,N , aˆ ,...,aˆ ) (A22) T † −k,1 −k,N X Y− Aˆ = P −1Bˆ = k k Bˆ , (A31) a Nambu space vector constructed from the Fourier- k k k T † k Yk X−k transformed HP bosons associated with the N basis sites in the magnetic unit cell. The 2N × 2N linear spin-wave kernel or elementwise N Mk Nk H = S (A23) = ˆ + ∗ ˆ† , k † T aˆk,α [(Xk )λαbk,λ (Y−k )λαb−k,λ] (A32a) N M− k k λ=1 has entries N † = ∗ ˆ† + ˆ . aˆ− ,α [(X−k )λαb− ,λ (Yk )λαbk,λ] (A32b) +− 1 k k = I − δ · z + Izz , λ= (Mk )μν k,μν μ,ν b eμ 0,μλ (A24a) 1 S λ To obtain Pk’s submatrices Xk and Yk we invoke the = I++ . (Nk )μν k,μν (A24b) following numerical recipe. We unitarily diagonalize UHk and then paranormalize the resulting eigenvectors, i.e., the We abbreviated − − columns of P 1, by enforcing Eq. (A29). With P 1 at hand, ξζ ξζ k k I ≡ I ik·(Δr+rν −rμ ), k,μν rμ,Δr+rν e (A25) Eq. (A31) is used to read off Xk and Yk. Δr

b. Anharmonic (cubic) theory

Before turning to Hˆ3, we note that spin-wave theory is an expansion about a (meta)stable state, that is, about a (local or global) energy minimum. Consequently, there are no forces that drive the system into a different state. In the second-quantized language this means that there is no spontaneous creation (decay) of a single particle out of (into) vacuum. Mathematically, this is expressed by √   S S −z † +z z− † z+ = ˆ = + I + I + + I + I 0 H1 aˆr rμ r+rμ,r +rν aˆr+rμ r+rμ,r +rν aˆr rν r+rμ,r +rν aˆr +rν r+rμ,r +rν 2 r,r rμ,rν √   + · − + † · + . S aˆr+rμ b eμ aˆr+rμ b eμ (A33) r rμ

033491-7 MOOK, KLINOVAJA, AND LOSS PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

This condition can be used to find the classical magnetic ground state. In our numerical setup, we directly minimize Hˆ0, which is an equivalent procedure. Numerically, the absence of Hˆ1 can be checked by the positivity and realness of the single-particle energies. The full cubic Hamiltonian after HP transformation reads √  S † −z † † +z ˆ =− + + I + + I H3 aˆr rμ aˆr +rν aˆr rν r+rμ,r +rν aˆr+rμ aˆr +rν aˆr rν r+rμ,r +rν 2 r,r rμ,rν ⎡√   † z− † † z+ 1⎣ S † −z + + + I + + I − + + I aˆr+rμ aˆr rμ aˆr rν r+rμ,r +rν aˆr+rμ aˆr rμ aˆr +rν r+rμ,r +rν aˆr+rμ aˆr rμ aˆr rμ r+rμ,r +rν 4 2 r,r rμ,rν  † † +z † z− † † z+ + + I + + + I + + I aˆr+rμ aˆr+rμ aˆr rμ r+rμ,r +rν aˆr +rν aˆr rν aˆr rν r+rμ,r +rν aˆr +rν ar +rν aˆr rν r+rμ,r +rν ⎤   + √1 † · − + † † · + ⎦. aˆr+rμ aˆr+rμ aˆr+rμ b eμ aˆr+rμ aˆr+rμ aˆr+rμ b eμ (A34) S r rμ

1 ˆ The term with the 4 factor in front has the same structure as H1 because within each term all bosonic operators belong to the same site. Consequently, if Hˆ1 vanishes, so does this part of Hˆ3. In the remaining part of Hˆ3, we move creators to the left of annihilators, √   S † † +z † † z+ ˆ =− + I + + I + . . . H3 aˆr+rμ aˆr +rν aˆr rν r+rμ,r +rν aˆr+rμ aˆr +rν aˆr rμ r+rμ,r +rν H c (A35) 2 r,r rμ,rν

We used that no spin interacts with itself, i.e., Ir,r = 0. After a Fourier transformation, Eq. (A35) assumes the form √   S † † +z † † z+ Hˆ3 =− √ aˆ aˆ ,ν aˆk+q,ν I ,μν + aˆ aˆ ,ν aˆk+q,μI− ,μν + H.c. . (A36) 2 N k,μ q k k,μ q q u k,q μ,ν μ ν Iz+ = Iz+ T = I+z We simplified notation: The sum over rμ and rν is denoted as a sum over and . One can show that −q,μν ( −q,μν ) q,νμ, T using the general property Ir,r = (Ir,r) of magnetic interaction matrices. Finally, we find the following generic form of the cubic Hamiltonian in terms of HP bosons: √ p=k+q S 1 λμ←ν † † Hˆ = √ V aˆ aˆ ,μaˆ ,ν + H.c. . (A37) 3 2!1! k,q←p k,λ q p Nu λ,μ,ν k,q,p λμ←ν =−δ I+z − δ I+z The interaction vertex Vk,q←p μν k,λμ λν q,μλ is a complex number in general; its expression agrees with that derived in Ref. [155]. The factorial factors are written to stress the symmetric of Hˆ3, i.e., creators and annihilators can be permuted among themselves. In terms of normal modes, Eqs. (A32a) and (A32b), the cubic Hamiltonian reads √ p=k+q S  ∗ ˆ = √ αβ←γ † † + αβ←γ † H3 Vk,q←p aˆk,αaˆq,β aˆ p,γ Vk,q←p aˆ p,γ aˆq,β aˆk,α 2 N α,β,γ k,q,p √ p=k+q S = √ αβ←γ ∗ ˆ† + ˆ ∗ ˆ† + ˆ ˆ + ∗ ˆ† Vk,q←p [(Xk )λαbk,λ (Y−k )λαb−k,λ][(Xq )μβ bq,μ (Y−q)μβ b−q,μ][(Xp)νγ bp,ν (Y−p)νγ b−p,ν ] 2 N α,β,γ λ,μ,ν k,q,p   + αβ←γ ∗ ∗ ˆ† + ˆ ˆ + ∗ ˆ† ˆ + ∗ ˆ† . Vk,q←p [(Xp)νγ bp,ν (Y−p)νγ b−p,ν ][(Xq)μβ bq,μ (Y−q)μβ b−q,μ][(Xk )λαbk,λ (Y−k )λαb−k,λ] (A38) ˆ = ˆ d + ˆ s Upon expansion, one finds that the cubic Hamiltonian assumes the form H3 H3 H3 , where the decay (d) part reads √ p=k+q S  ∗ ˆ d = √ αβ←γ ∗ ∗ ˆ† ˆ† ˆ + αβ←γ ∗ ∗ ˆ† ˆ† ˆ H3 Vk,q←p (Xk )λα (Xq )μβ (Xp)νγ bk,λbq,μbp,ν Vk,q←p (Y−k )λα (Y−q)μβ (Y−p)νγ b−k,λb−q,μb−p,ν 2 N α,β,γ λ,μ,ν k,q,p   + αβ←γ ∗ ∗ ˆ† ˆ† ˆ + αβ←γ ∗ ∗ ∗ ˆ† ˆ† ˆ Vk,q←p (Xk )λα (Y−q)μβ (Y−p)νγ b−p,ν bk,λb−q,μ Vk,q←p (Y−k )λα (Xq)μβ (Xp)νγ bp,ν b−k,λbq,μ   + αβ←γ ∗ ∗ ˆ† ˆ† ˆ αβ←γ ∗ ∗ ∗ ˆ† ˆ† ˆ + . . Vk,q←p (Y−k )λα (Xq )μβ (Y−p)νγ bq,μb−p,ν b−k,λ Vk,q←p (Xk )λα (Y−q)μβ (Xp)νγ b−q,μbp,ν bk,λ H c (A39)

033491-8 QUANTUM DAMPING OF SKYRMION CRYSTAL … PHYSICAL REVIEW RESEARCH 2, 033491 (2020) and the source (s) part √ p=k+q S ˆ s = √ αβ←γ ∗ ∗ ∗ ˆ† ˆ† ˆ† H3 Vk,q←p (Xk )λα (Xq )μβ (Y−p)νγ bk,λbq,μb−p,ν 2 N α,β,γ λ,μ,ν k,q,p   + αβ←γ ∗ ∗ ∗ ∗ ˆ† ˆ† ˆ† + . . . Vk,q←p (Xp)νγ (Y−q)μβ (Y−k )λαbp,ν b−q,μb−k,λ H c (A40) We now reverse the sign of the momenta in every second term to obtain √ p=k+q S ˆ d = √ αβ←γ ∗ ∗ + αβ←γ ∗ ∗ ∗ ˆ† ˆ† ˆ H3 Vk,q←p (Xk )λα (Xq )μβ (Xp)νγ (V−k,−q←−p) (Yk )λα (Yq )μβ (Yp)νγ bk,λbq,μbp,ν 2 N α,β,γ λ,μ,ν k,q,p + αβ←γ ∗ ∗ + αβ←γ ∗ ∗ ∗ ˆ† ˆ† ˆ Vk,q←p (Xk )λα (Y−q)μβ (Y−p)νγ (V−k,−q←−p) (Yk )λα (X−q)μβ (X−p)νγ b−p,ν bk,λb−q,μ + αβ←γ ∗ ∗ + αβ←γ ∗ ∗ ∗ ˆ† ˆ† ˆ + . . Vk,q←p (Y−k )λα (Xq )μβ (Y−p)νγ (V−k,−q←−p) (X−k )λα (Yq )μβ (X−p)νγ bq,μb−p,ν b−k,λ H c (A41) and √ p=k+q S ˆ s = √ αβ←γ ∗ ∗ ∗ + αβ←γ ∗ ∗ ∗ ∗ ˆ† ˆ† ˆ† + . . . H3 Vk,q←p (Xk )λα (Xq )μβ (Y−p)νγ (V−k,−q←−p) (X−p)νγ (Yq )μβ (Yk )λα bk,λbq,μb−p,ν H c 2 N α,β,γ λ,μ,ν k,q,p (A42) Finally, for symmetrization, we relabel momenta and band indices in the second and third lines of Eq. (A41). Using ν → λ → μ → ν and −p → k → q →−p in the second line and μ → λ → ν → μ and q → k →−p → q in the third line, we arrive at √ p=k+q d S 1 λμ←ν † † Hˆ = √ V bˆ bˆ ,μbˆ ,ν + H.c. , (A43) 3 2!1! k,q←p k,λ q p N λ,μ,ν k,q,p with the decay vertex   Vλμ←ν = αβ←γ ∗ ∗ + αβ←γ ∗ ∗ ∗ + αβ←γ ∗ ∗ k,q←p Vk,q←p (Xk )λα (Xq )μβ (Xp)νγ V−k,−q←−p (Yk )λα (Yq )μβ (Yp)νγ Vq,−p←−k(Xq )μα (Yp)νβ(Yk )λγ α,β,γ   + αβ←γ ∗ ∗ ∗ + αβ←γ ∗ ∗ + αβ←γ ∗ ∗ ∗ . V−q,p←k (Yq )μα (Xp)νβ(Xk )λγ V−p,k←−q(Yp)να(Xk )λβ (Yq )μγ (Vp,−k←q) (Xp)να(Yk )λβ (Xq )μγ (A44) Similarly, in Eq. (A42), we add the summand another two times, divide by 3, and cyclically permute momenta and band indices. We arrive at √ −p=k+q s S 1 λμν † † † Hˆ = √ W bˆ bˆ ,μbˆ ,ν + H.c. , (A45) 3 3!0! k,q,p k,λ q p N λ,μ,ν k,q,p with the source vertex   Wλμν = αβ←γ ∗ ∗ ∗ + αβ←γ ∗ ∗ ∗ ∗ + αβ←γ ∗ ∗ ∗ k,q,p Vk,q←p (Xk )λα (Xq )μβ (Yp )νγ V−k,−q←−p (Yk )λα (Yq )μβ (Xp)νγ Vq,p←k (Xq )μα (Xp)νβ(Yk )λγ α,β,γ     + αβ←γ ∗ ∗ ∗ ∗ + αβ←γ ∗ ∗ ∗ + αβ←γ ∗ ∗ ∗ ∗ . V−q,−p←−k (Yq )μα (Yp )νβ(Xk )λγ Vp,k←q (Xp)να(Xk )λβ (Yq )μγ V−p,−k←−q (Yp )να(Yk )λβ (Xq )μγ (A46) Again, we explicitly denoted the factorial factors, arising from symmetrization under permutation of creators (or annihilators) among themselves.

3. Many-body perturbation theory up to order 1/S

Up to second-order many-body perturbation theory, the normal one-magnon temperature Green’s function Gk,νν (τ ) = −T τ †  τ bk,ν ( )bk,ν reads [156] h¯β h¯β h¯β (0) 1 † con 1 † con ,νν τ ≈ τ + τ Tτ ˆ τ ˆ ,ν τ ˆ  − τ τ Tτ ˆ τ ˆ τ ˆ ,ν τ ˆ  , Gk ( ) Gk,νν ( ) d 1 H4( 1)bk ( )bk,ν (0) 2 d 1 d 2 H3( 1)H3( 2 )bk ( )bk,ν (0) h¯ 0 2!¯h 0 0 (A47) (0) τ τ T with Gk,νν ( ) the bare propagator. The ’s denote imaginary time, τ is the “time”-ordering operator, the subscript (0) denotes −1 averaging with respect to Hˆ2, and “con” refers to connected diagrams; β = (kBT ) , where kB is Boltzmann’s constant and / (0) τ T temperature. Both the first-order and the second-order term are 1 S corrections to Gk,νν ( ). Note that a second-order term

033491-9 MOOK, KLINOVAJA, AND LOSS PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

FIG. 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 (V) and source (W) three-magnon vertices (or their conjugates) given in Eqs. (A44)and(A46), respectively. (a) First-order Hartree-Fock corrections due to Hˆ4 that yield a static real self-energy. (b) Second-order tadpole diagrams due to Hˆ3 that also yield a static real self-energy. (c) Second-order bubble diagrams due to Hˆ3 that yield a dynamic self-energy. However, only the first two diagrams contribute to the imaginary part of the self-energy.

2 in Hˆ4 would yield a 1/S correction and is therefore dropped. [Even for S  1, spin-wave theory is surprisingly reliable and higher-order terms may be dropped because the expansion in Eq. (A17) is also effectively an expansion in the magnon number a†a, which is small at low temperatures.] By virtue of Wick’s theorem, one shows that the first-order Hˆ4 term gives rise to frequency-independent Hartree-Fock corrections [cf. Fig. 7(a)] that cause a real self-energy [156,157]. Similarly, the second-order Hˆ3 term gives rise to frequency- independent tadpole diagrams [cf. Fig. 7(b)] that also contribute a real self-energy [157]. Being interested in magnon damping, which is associated with the imaginary part of the self-energy, we drop all frequency-independent 1/S corrections. Out of the remaining bubble diagrams [cf. Fig. 7(c)], which also derive from Hˆ3, only the first two yield complex diagonal normal self-energies (ν = ν): ρ ε + ρ ε + ρ ε − ρ ε Σ ε = 1 1Vλμ←ν 2 ( q,λ) ( k−q,μ) 1 + Vνμ←λ 2 ( q,μ) ( k+q,λ) . k,ν ( ) q,k−q←k + k,q←k+q + (A48) N 2 ε + i0 − ε ,λ − ε − ,μ ε + i0 + ε ,μ − ε + ,λ u q λ,μ q k q q k q ρ = βx − −1 Wλμν Here (x) (e 1) is the Planck distribution. Note that the source vertex k,q,p does not enter Eq. (A48) because the source-sink bubble [right diagram of Fig. 7(c)] is off-resonance and we dropped it. Within the on-shell approximation ε → εk,ν , the magnon damping rate is given by k,ν ≡−Im[Σk,ν (εk,ν )]. Ignoring the real part of the self-energy, the normal one- −1 ε ≈ ε − ε +  ε ≈ magnon (retarded) Green’s function reads Gk,ν ( ) k,ν i k,ν , from which we obtain the spectral function Ak,ν ( ) −ImGk,ν (ε)/π of band ν. It follows from the Sokhotski-Plemelj theorem that the temperature-dependent damping reads π λμ←ν 2  ,ν (T ) = V δ(ε ,ν − ε ,λ − ε − ,μ)[ρ(ε ,λ, T ) + ρ(ε − ,μ, T ) + 1] k 2N q,k−q←k k q k q q k q u q λ,μ π νμ←λ 2 + V δ(ε ,ν + ε ,μ − ε + ,λ)[ρ(ε ,μ, T ) − ρ(ε + ,λ, T )]. (A49) N k,q←k+q k q k q q k q u q λ,μ spon The first line in Eq. (A49) comprises magnon decay. The spontaneous damping k,ν in Eq. (2) is obtained by taking the limit T → 0. The second line in Eq. (A49) is a two-magnon collision. It takes finite temperature to have thermal magnons available for collisions, which is why this term is frozen out as T → 0.

APPENDIX B: ADDITIONAL RESULTS Fig. 9 we use a data representation similar to that in Fig. 4 (histograms viewed from above and square color and size 1. Extended Fig. 2 denoting the same information). Figure 8 extends Fig. 2.InFigs.8(a)–8(g) the pronounced Two additional conclusions can be drawn. shift of the flat modes due to the Zeeman energy can be First, note that the symbol × in Fig. 9 marks for- noticed. The two-magnon DOS, shown in Figs. 8(h)–8(n), bidden decay channels. (Numerically, we found that they inherits this field dependence, as revealed by the field-tunable have amplitudes less than 10−22 meV2, which is negligi- sharp yellow features. The overlap of the single-particle bands ble compared to other weak channels with amplitudes of with regions of a large two-magnon DOS leads to strong the order of 10−4 meV2.) Thus, decay channels with identi- quasiparticle decay and lifetime band broadening in the spec- cal decay products are inactive for the G, A, T, C, ν = 9, tral function [Figs. 8(o)–8(u)]. Several instances are marked and P modes. We conclude that these modes cannot decay by red arrows. into a single flat mode at half their energy. Note that this result is only correct for k = 0 excitations. (We numeri- cally verified that all decay channels are possible for k = 0 = 2. Analysis of 1000 decay channels at k 0, involving excitations.) the ten lowest magnon modes Second, the decay efficiencies of polygon modes [ν = 1, In the main text, we presented a decay channel analysis of 2, 4, 7, and 10 for the G, E, T, Q, and P mode in Figs. 9(a), the ABC modes. Here we extend this analysis to the lowest ten 9(b), 9(d), 9(g), and 9(j), respectively] are overall larger than magnon bands, resulting in 1000 possible decay channels. In those of the ABC modes. Moreover, we find particularly large

033491-10 QUANTUM DAMPING OF SKYRMION CRYSTAL … PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

FIG. 8. Same as Fig. 2 but for additional magnetic field values.

Cλμ←ν for decays that conserve l, e.g., 3. Decay channels analysis of ABC modes dependent on the magnetic field

10 → (2, 4), l = 5 = 2 + 3 [Fig. 9(j), P mode], Here we study the magnetic-field dependence of the decay channels of the ABC modes. Results are presented in Fig. 10. 10 → (1, 7), l = 5 = 1 + 4 [Fig. 9(j), P mode], As far as the mode labeling is concerned, recall that the flat 7 → (2, 2), l = 4 = 2 + 2 [Fig. 9(g), Q mode], polygon modes overtake the dispersive modes as the field increases, which is why the labels in Fig. 10 are different for 7 → (1, 4), l = 4 = 1 + 3 [Fig. 9(g), Q mode], different fields. 4 → (1, 2), l = 3 = 1 + 2 [Fig. 9(d), T mode], The field increases going from left to right. The instability of the C mode [Fig. 10(a)] towards the channel C →{A, G} increases, while that towards the C →{E, G} channel de- which may be understood as a kind of “weak” selection rule. creases. Other decay channels exhibit a weaker increase. However, since there is no conservation law for l, processes For the B mode [Fig. 10(b)] we find that channels towards that increase or decrease l also have nonzero amplitude; see, lower-energy modes (G, E, and A) become less efficient at e.g., Fig. 9(b), E mode, 2 → (3, 4), for which l increases. higher fields, resulting in an effective field-enhanced stability This nonconservation is attributed to broken rotational of the B mode, although decay channels towards higher- symmetry. While an isolated skyrmion (with size much larger energy modes increase considerably. than the lattice spacing) is rotationally symmetric, which Finally, we find that almost all decay channels of the A translates into conservation of azimuthal mode number l mode [Fig. 10(c)] get frozen out by the field. This complies similar to what was found for magnetic vortices [158], the with the finding that the A mode is the most stable mode skyrmions in a SkX experience a hexagonal deformation. among the ABC modes. Notice in particular that the decay With the necessary continuous symmetry broken, there is no channels towards lower-energy modes (G and E modes) are conserved quantity. always negligible, even at low fields.

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(a) ν = 1: gyrotropic (G, l =1) (b) ν = 2: elliptic (E, l =2) (c) ν = 3: anticlockwise (A) (d) ν = 4: triangular (T, l =3) (e) ν = 5: breathing (B) μ G E A T B C Q P G E A T B C Q P G E A T B C Q P G E A T B C Q P G E A T B C Q P λ 1 2 3 4 5 6 7 8910 1 2 3 4 5 6 7 8910 1 2 3 4 5 6 7 8910 1 2 3 4 5 6 7 8910 1 2 3 4 5 6 7 8910 G 1 × × × E 2 × × × A 3 × × × T 4 × × × B 5 × × × C 6 × × × Q 7 × × × 8 × × × 9 × × × P 10 × × ×

(f) ν = 6: clockwise (C) (g) ν = 7: quadrupolar (Q, l =4) (h) ν =8 (i) ν =9 (j) ν = 10: pentagonal (P, l =5) μ G E A T B C Q P G E A T B C Q P G E A T B C Q P G E A T B C Q P G E A T B C Q P λ 1 2 3 4 5 6 7 8910 1 2 3 4 5 6 7 8910 1 2 3 4 5 6 7 8910 1 2 3 4 5 6 7 8910 1 2 3 4 5 6 7 8910 G 1 × × × E 2 × × × A 3 × × × T 4 × × × B 5 × × × C 6 × × × Q 7 × × × 8 × × × 9 × × × P 10 × × ×

FIG. 9. Decay channel analysis of an M = 8SkXatb/JS = 0.5andk = 0. (a)–(j) Decay channel efficiency Cλμ←ν for λ, μ, ν = 1,...,10: (a) ν = 1, gyrotropic mode, l = 1; (b) ν = 2, elliptic mode, l = 2; (c) ν = 3, anticlockwise mode; (d) ν = 4, triangular mode, l = 3; (e) ν = 5, breathing mode; (f) ν = 6, clockwise mode; (g) ν = 7, quadrupolar mode, l = 4; (h) ν = 8; (i) ν = 9; and (j) ν = 10, pentagonal mode, l = 5. Small blue/large red squares indicate small/large Cλμ←ν . The symbol × indicates exactly zero Cλμ←ν . Notice the decay product symmetry λ ↔ μ.

4. Effects of finite temperature on magnon damping [ρ(εq,λ, T ) + ρ(ε−q,μ, T )] in Eq. (B2). If one of the two de- At finite T , one obtains not only additional thermal weight- cay products happens to be the G mode, which is lowest in ing of the decay processes [first line in Eq. (A49)], but also energy, the thermal Bose weight is large, especially for small additional collision processes (second line). Here we evalu- q. However, if one of the decay products has low energy and ate the temperature dependence of the damping of the ABC low momentum, the energy and momentum of the other decay modes (ν = 3, 5, 6atb/JS = 0.485). We decompose the total product have to be close to those of the decaying magnon, that damping as is, within the same mode (but necessarily at lower energy). Such a decay process is sketched in Fig. 11(d). Reviewing the total = spon + decay + coll , 0,ν (T ) 0,ν 0,ν (T ) 0,ν (T ) (B1) energy dispersion of the ABC modes, one finds that only that of the A mode is concave at the Brillouin zone center. Hence, where the spontaneous part spon is what we have discussed 0,ν the aforementioned type of thermally enhanced decay process so far, can only happen for the A mode. π decay λμ←ν 2 coll  (T ) = V δ(ε ,ν − ε ,λ − ε− ,μ) The scenario for damping due to collisions 0,ν (T ), shown 0,ν 2N q,−q←0 0 q q u q λ,μ in Fig. 11(c), is opposite; the A mode exhibits the lowest rela- tive increase. The factor [ρ(εq,μ, T ) − ρ(εq,λ, T )] in Eq. (B3) × [ρ(ε ,λ, T ) + ρ(ε− ,μ, T )] (B2) q q tells us that a low-energy magnon at εq,μ, which collides ε ε is damping due to thermally activated decay processes, and with the magnon at 0,ν to form a q,λ magnon, dominates π the thermal weight. If this low-energy magnon is in the G coll νμ←λ2  (T ) = V δ(ε ,ν + ε ,μ − ε ,λ) mode, its contribution in energy and momentum to the final 0,ν N 0,q←q 0 q q u q λ,μ magnon is small. Thus, the final magnon must have energy and momentum close to the ε0,ν magnon (but necessarily × [ρ(ε ,μ, T ) − ρ(ε ,λ, T )] (B3) q q higher energy). Such states are only available for the convex is damping due to thermally activated collisions. B and C modes, as indicated in Fig. 11(e), but absent for the We choose a field b/JS = 0.485, for which the sponta- concave A mode. neous damping of the C mode is maximal [cf. Fig. 3(c)]. Combined, thermal damping and collision processes lead Results for the total damping dependent on temperature are to the rather similar temperature trend of the ABC modes’ shown in Fig. 11(a). As expected, finite temperatures lead to damping shown in Fig. 11(a). The takeaway message from an increase of magnon damping. this analysis is that the thermally enhanced damping of the C As shown in Fig. 11(b), the A mode (blue line) exhibits mode at kBT/JS = 0.2 is still a factor of 2 smaller than its the largest relative increase in damping due to thermally acti- spontaneous damping at T = 0 [cf. green line in Fig. 11(a)]. decay vated decays, 0,ν (T ). This can be explained by the factor We conclude that the quantum damping is also observable for

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Magnetic field

b = 0.2/(JS) b = 0.4/(JS) b = 0.6/(JS) b = 0.8/(JS)

(a) clockwise mode (C) (a) anticlockwise mode (A) (a) anticlockwise mode (A) (a) anticlockwise mode (A) μ G E A T B Q C P μ G E A T B C Q P μ G E A T B C Q P μ G E A B T C Q λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 78910 G 1 × G 1 × G 1 × G 1 × E 2 × E 2 × E 2 × E 2 × A 3 × A 3 × A 3 × A 3 × T 4 × T 4 × T 4 × B 4 × B 5 × B 5 × B 5 × T 5 × Q 6 × C 6 × C 6 × C 6 × C 7 × Q 7 × Q 7 × 7 × 8 × 8 × 8 × Q 8 × 9 × 9 × 9 × 9 × P 10 × P 10 × P 10 × P 10 ×

(b) breathing mode (B) (a) anticlockwise mode (A) (a) anticlockwise mode (A) (a) anticlockwise mode (A) μ G E A T B Q C P μ G E A T B C Q P μ G E A T B C Q P μ G E A B T C Q λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 78910 G 1 G 1 G 1 G 1 E 2 E 2 E 2 E 2 A 3 A 3 A 3 A 3 T 4 T 4 T 4 B 4 B 5 B 5 B 5 T 5 Q 6 C 6 C 6 C 6 C 7 Q 7 Q 7 7 8 8 8 Q 8 9 9 9 9 P 10 P 10 P 10 P 10

(c) anticlockwise mode (A) (a) anticlockwise mode (A) (a) anticlockwise mode (A) (a) anticlockwise mode (A) μ G E A T B Q C P μ G E A T B C Q P μ G E A T B C Q P μ G E A B T C Q λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 7 8910 λ 1 2 3 4 5 6 78910 G 1 × G 1 × G 1 × G 1 × E 2 × E 2 × E 2 × E 2 × A 3 × A 3 × A 3 × A 3 × T 4 × T 4 × T 4 × B 4 × B 5 × B 5 × B 5 × T 5 × Q 6 × C 6 × C 6 × C 6 × C 7 × Q 7 × Q 7 × 7 × 8 × 8 × 8 × Q 8 × 9 × 9 × 9 × 9 × P 10 × P 10 × P 10 × P 10 ×

FIG. 10. Decay channel efficiencies Cλμ←ν 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 Cλμ←ν (large red squares).

finite-temperature SkXs, whose stability relies on moderate form thermal fluctuations. M(T ) ≈ M0 + δM0 + M2(T ). (B4)

5. Consistency of spin-wave theory −1 Here M0 =−V ∂E0/∂b is the classical ground-state mag- The Mermin-Wagner theorem states that two-dimensional netization obtained from the classical ground-state energy E0 magnets with spin space isotropy exhibit no long-range order (or Hˆ0) and V is the sample’s volume. The lowest-order quan- −1 at nonzero temperatures [159]. Within spin-wave theory, as tum correction to the magnetization δM0 =−V ∂δE0/∂b δ = pioneered by Bloch [160], this is signaled by a diverging is associated with the harmonic zero-point energy E0 ε − / magnetization M(T )forT > 0, indicating the failure of the ν,k[ k,ν (Mk )νν] 2, with Mk given in Eq. (A24a). Fi- =− −1 ρ ε , ∂ε /∂ initial assumption of an ordered ground state. nally, M2(T ) V ν,k ( k,ν T ) k,ν b accounts for For a chiral magnet, spin space isotropy is broken due to the thermal occupation of magnons. Consistency demands DM interaction and the Mermin-Wagner theorem does not that neither δM0 nor M2(T ) diverges, i.e., neither quantum nor apply. Nonetheless, we need to check for consistency to val- thermal fluctuations destroy magnetic order. idate our spin-wave theory approach. We do so by noting Let us quickly review the failure of spin-wave theory for a that up to harmonic order the magnetization assumes the two-dimensional ferromagnet with spin space isotropy [160].

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anticlockwise

= / = . total = spon + FIG. 11. Damping of ABC modes of an M 8SkXatb JS 0 485 dependent on temperature T . (a) Total damping 0 (T ) 0 decay + coll decay coll 0 (T ) 0 (T ), (b) damping 0 (T ) due to thermally activated decays, and (c) damping 0 (T ) 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.

2 Since there are no quantum fluctuations (δM0 = 0), only low momenta, we find that is goes as k [Fig. 12(b)]. Conse- quently, the divergence of the magnetization is lifted: 1 ∝ 2   M2(T ) ε / d k (B5) 1 ∂εk e k,ν (kBT ) − 1 ∝ ∝ < ∞. n BZ M2(T ) T dk T kdk (B7) 0 k ∂b 0 remains to be investigated. Integration is over the first Bril- Since typical chiral magnets have small D/J ratios, the −∂ε /∂ louin zone. (We used that k,ν b is constant because all crucial ingredient to break spin space isotropy is also small. magnons carry the same nonzero magnetic moment.) The This installs a threshold temperature above which spin-wave ε ∝ 2 lowest-energy magnon with energy k,1 k is a Goldstone theory breaks down. The first almost flat magnon mode, which > =| | < mode. Thus, for T 0, there is a region k k for may exhibit a divergent thermal contribution to the magneti- which εk,1 kBT (>0 being an upper cutoff). Conse- 2 zation, approximately appears at an energy ε0 = JS(D/J) . quently, after expanding the exponential in Eq. (B5), we find Due to the almost constant magnetic moment of flat modes, its thermal contribution to M(T ) reads   ∝ k ∝ 1 M2(T ) T dk T dk (B6) ∝ ρ ε , 2 . 0 εk,1 0 k M2(T ) ( 0 T ) d k (B8) BZ and encounter the divergence of thermal fluctuations. In the limit of D/J 1 and ε0 kBT ,wehaveρ(ε0, T ) ∝ kBT 2 ∝ / 2 For a SkX in the continuous limit, there is also a Goldstone JS(D/J)2 and the Brillouin zone area BZ d k (D J) ; hence 2 mode with εk ∝ k . It is associated with the spontaneously kBT broken translational invariance. Hence, it costs no energy to M2(T ) ∝ . (B9) rigidly translate the SkX in space. In our discrete simulations, JS translational invariance is already broken on the Hamiltonian Notice that the D/J ratio cancels out. Thus, spin-wave theory level and the Goldstone mode becomes the gyrotropic mode for a two-dimensional SkX is consistent if with a tiny spin-wave gap. Nonetheless, we can extract its kBT JS. (B10) magnetic moment −∂εk,1/∂b, as depicted in Fig. 12(a).For Since kBTc ∼ JS is approximately the ordering temperature of the SkX [116], Eq. (B10) tells us that validity of (harmonic) spin-wave theory is restricted to temperatures well below the ordering temperature, which is the usual realm of applicabil- ity. At T = 0, we are still concerned with quantum fluctuations 1 ∂ε ,ν ∂(M )νν δM =− k − k (B11) 0 2V ∂b ∂b ν,k due to the noncollinearity of the SkX. Note that there are no divergent contributions in Eq. (B11). This is verified by direct numerical evaluation. In Fig. 13 we show the ratio δM0/SM0, FIG. 12. Magnetic moment of a SkX’s translational Goldstone encoding the relative importance of quantum corrections with mode (a) along high-symmetry paths and (b) dependent on the dis- respect to the classical ground-state magnetization M0, depen- tance from the Brillouin zone center. dent on the skyrmion unit cell size. Even for tiny skyrmion

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the averaged three-magnon vertex total total 2 λμ←ν 1 V0,0←0 1 V0,0←0 D V , ← ≡ ≈   ∝ J 0 0 0 6 N 6 π/ D 2 J 2 arctan J (B14) scales with J(D/J)2 if D/J 1. The same finding applies to finite momenta, which merely introduce phase factors. (The 1 factor 6 is due to the six nearest neighbors on the triangular lattice.) In the limit of large skyrmions, the degree of noncollinear- ity is tiny, which means for the spin-wave kernel Hk in Eq. (A23) that the anomalous coupling matrix Nk ∝ D/J is tiny. Consequently, the Y ∝ D/J submatrix of P in FIG. 13. Ratio of quantum corrections to the magnetization δM k k 0 Eq. (A31) is tiny too. As for the decay vertex in Eq. (A44), and the classical ground-state magnetization M0 dependent on D/J (or linear skyrmion size M). We set b/JS = 0.5(D/J)2. we conclude that on average 2 λμ←ν αβ←γ ∗ ∗ D V ≈ V (X )λα (X )μβ (X )νγ ∝ J = k,q←k+q k,q←k+q k q p unit cells with a linear size of M 5 (25 spins per unit α,β,γ J cell), the quantum mechanical magnetization corrections are (B15) smaller than 3% (for S = 1). We conclude that quantum fluc- 0 tuations are negligible. is the dominating contribution because Xk ∝ (D/J) .Intro- ducing the mean decay amplitude as 6. Dependence of quantum damping on skyrmion size 4 1 1 λμ←ν D Vν 2 ≡ V 2 ∝ J2 (B16) In Fig. 14 we plot the total three-magnon vertex k 2 k,q←k+q Nu N J   q λ,μ total λμ←ν +z +z V ≡ V =− δμν I + δλν I k,q←k+q k,q←k+q k,λμ q,μλ (note that the sum over λ and μ and the 1/N2 factor have λ,μ,ν λ,μ,ν inverse scaling with D/J such that their influence cancels out), (B12) we find for the spontaneous damping that = = per magnetic unit cell at zero momenta (k q 0), i.e., π D 4 D spon ≈ Vν 2D2m ε ∝ → k,ν k k ( ν,k ) J as 0 (B17) total =− I+z , 2 J J V0,0←0 2 0,μν (B13) μ,ν D2m ∝ / because k 1 J.WeuseEq.(B17) to estimate the damp- spon = , , dependent on the D/J ratio (skyrmion size). As D/J → 0, a ing of the ABC modes X , with X A B C. As shown in spon ≈ = −3 = −2 = single skyrmion becomes infinitely large, but the total three- Fig. 3(c), X FX J, with FA 10 , FB 10 , and FC −1 / = spon ≈ / 4 total 10 (for D J 1). Hence, we expect X FX J(D J) . magnon vertex V0,0←0 converges to a nonzero constant. Thus,

7. Influence of dimensionality on magnon damping As mentioned in the main text, the zero-temperature damp- ing is directly related to the two-magnon density of states, which derives from the δ function in Eq. (A49) (first sum). In two dimensions, the onset of the two-magnon DOS is a Heaviside step function. Thus, as a single-particle band crosses the onset of the two-magnon DOS, its damping also exhibits a step, which renders the magnon damping and linewidth broadening discontinuous. In contrast, in three di- mensions, the onset is given by a square root (see also Refs. [95,97]). Hence, if instead of a two-dimensional SkX we studied a three-dimensional SkX, e.g., built from stack- ing SkX layers, we would expect a renormalization of the step. However, this renormalization is not as drastic as that associated with going from one to two dimensions, for which the Van Hove singularity of the two-magnon DOS changes from an inverse square root to the step function [97]. Thus, in general, the overall magnitude of the sponta- FIG. 14. Integrated three-magnon vertex dependent on D/J neous magnon damping is not expected to be considerably (bottom abscissa) and linear skyrmion size M (top abscissa). We set smaller in three than in two dimensions. Hence, details be- b/JS = 0.5(D/J)2. come important, above all the magnetic texture along the third

033491-15 MOOK, KLINOVAJA, AND LOSS PHYSICAL REVIEW RESEARCH 2, 033491 (2020) dimension. For example, skyrmion tubes, which are stacks of we approximate by setting j = 1. For the classical Gilbert α ≈ α j ≈ αε2D two-dimensional skyrmion crystals, are expected to exhibit damping, the magnon damping reads n, j En n . spontaneous decay different from that of Bloch point crystals, The associated critical thickness is thus which are genuinely three-dimensional noncollinear textures. For a noncollinear order along the third direction, larger con- SJ⊥ L ≈ π d. (B19) tributions to the three-magnon vertices are expected than for a class αε2D collinear order. n There is another important aspect of three-dimensional Taking the C mode as reference, we have ε2D ≈ JS(D/J)2. SkXs, associated with the magnon bands acquiring dispersion n With J⊥ = J, as expected for a cubic crystal, it follows that along the third direction [161]. For quasi-two-dimensional layered structures with weak interlayer coupling much smaller π J than the size of typical magnon band gaps in two dimensions, L ≈ √ d, (B20) class α D the dispersion along the additional dimension is negligible. However, for interlayer coupling as large as (or larger than) which amounts to L ∼ 5000d for parameters of magnon band gaps, the flat bands in two dimensions can − class Cu OSeO (α = 10 4 and D/J ≈ 0.06, taken from no longer be considered flat and their influence on magnon 2 3 Refs. [138] and [8], respectively). In contrast, for the damping becomes less abrupt. spon spontaneous quantum damping, we set  ≈ 0.1J(D/J)4 For a real sample of finite thickness L, we expect a C (see Appendix B6) and find transition between two-dimensional and three-dimensional behavior with increasing thickness. The finite thickness acts 2 10SJ⊥ J as a quantum well, introducing several magnon energy levels L ≈ π d. (B21) spon J D 2π 2 2 j 2D j h¯ E = ε + , (B18) n n ∗ 2 = = ∼ 2m L With S 1 and J⊥ J, we obtain Lspon 2700d for with j for each magnon band n with energy Cu2OSeO3. In terms of the relative importance of quantum ε2D to classical damping RC, as introduced in Sec. IV, the relation n in two dimensions. We assumed a parabolic dispersion along the third direction, ignoring subleading nonreciprocal  = shifts due to DMI. The respective effective magnon mass Lclass RCLspon (B22) ∗ 2 2 m = h¯ /2SJ⊥d includes the exchange interaction J⊥ along is extracted. Obviously, if the quantum damping is stronger j the third direction and the layer spacing d.TheEn ’s become than the classical damping, it sets the critical thickness for the dense as L →∞and periodic boundary conditions may be crossover between two-dimensional and three-dimensional applied. A critical thickness L may be defined by the con- behavior. dition that the broadening of the discrete magnon quantum We conclude that the results on spontaneous decay we have well levels, due to either spontaneous decay or Gilbert damp- obtained in the strictly two-dimensional limit apply to thin ≈ π / 2 , ing, equals their energetic spacing E SJ⊥( d L) , which films with a thickness L min(Lclass Lspon ).

[1] A. N. Bogdanov and D. Yablonskii, Zh. Eksp. Teor. Fiz. 95, K. Yanai, H. Nakamura, D. Ehlers, V. Tsurkan, and A. Loidl, 178 (1989). Nat. Mater. 14, 1116 (2015). [2] A. N. Bogdanov and U. K. Rößler, Phys. Rev. Lett. 87, 037203 [10] G. Finocchio, F. Büttner, R. Tomasello, M. Carpentieri, and (2001). M. Kläui, J. Phys. D 49, 423001 (2016). [3] U. K. Rößler, A. N. Bogdanov, and C. Pfleiderer, Nature [11] S. Seki and M. Mochizuki, Skyrmions in Magnetic Materials (London) 442, 797 (2006). (Springer International, Cham, 2016). [4] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. [12] A. Soumyanarayanan, M. Raju, A. L. Gonzalez Oyarce, Neubauer, R. Georgii, and P. Böni, Science 323, 915 (2009). A. K. C. Tan, M.-Y. Im, A. P. Petrovic,´ P. Ho, K. H. Khoo, M. [5] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Tran, C. K. Gan, F. Ernult, and C. Panagopoulos, Nat. Mater. Matsui, N. Nagaosa, and Y. Tokura, Nature (London) 465, 901 16, 898 (2017). (2010). [13] A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo, [6] W. Münzer, A. Neubauer, T. Adams, S. Mühlbauer, C. Franz, F. Damay, U. K. Rößler, C. Felser, and S. S. P. Parkin, Nature F. Jonietz, R. Georgii, P. Böni, B. Pedersen, M. Schmidt, A. (London) 548, 561 (2017). Rosch, and C. Pfleiderer, Phys. Rev. B 81, 041203(R) (2010). [14] T. Kurumaji, T. Nakajima, V. Ukleev, A. Feoktystov, T.-h. [7] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Arima, K. Kakurai, and Y. Tokura, Phys. Rev. Lett. 119, Ishiwata, Y. Matsui, and Y. Tokura, Nat. Mater. 10, 106 (2011). 237201 (2017). [8] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 [15]A.Fert,N.Reyren,andV.Cros,Nat. Rev. Mater. 2, 17031 (2012). (2017). [9] I. Kézsmárki, S. Bordács, P. Milde, E. Neuber, L. M. Eng, [16] W. Jiang, G. Chen, K. Liu, J. Zang, S. G. te Velthuis, and A. J. S. White, H. M. Rønnow, C. D. Dewhurst, M. Mochizuki, Hoffmann, Phys. Rep. 704, 1 (2017).

033491-16 QUANTUM DAMPING OF SKYRMION CRYSTAL … PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

[17] Y. Zhou, Natl. Sci. Rev. 6, 210 (2018). [45] P. M. Buhl, F. Freimuth, S. Blügel, and Y. Mokrousov, Phys. [18] K. Everschor-Sitte, J. Masell, R. M. Reeve, and M. Kläui, Status Solidi 11, 1700007 (2017). J. Appl. Phys. 124, 240901 (2018). [46] B. Göbel, A. Mook, J. Henk, and I. Mertig, Phys. Rev. B 96, [19] B. Binz and A. Vishwanath, Physica B 403, 1336 (2008). 060406(R) (2017). [20] M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong, Phys. [47] X. Zhang, J. Müller, J. Xia, M. Garst, X. Liu, and Y. Zhou, Rev. Lett. 102, 186601 (2009). New J. Phys. 19, 065001 (2017). [21] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. [48] M. Garst, J. Waizner, and D. Grundler, J. Phys. D 50, 293002 Niklowitz, and P. Böni, Phys.Rev.Lett.102, 186602 (2009). (2017). [22] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, [49] S. A. Díaz, J. Klinovaja, and D. Loss, Phys. Rev. Lett. 122, C. Pfleiderer, K. Everschor, M. Garst, and A. Rosch, Nat. 187203 (2019). Phys. 8, 301 (2012). [50] S.-Z. Lin, J.-X. Zhu, and A. Saxena, Phys. Rev. B 99, [23] M. Mochizuki, Phys. Rev. Lett. 108, 017601 (2012). 140408(R) (2019). [24] Y. Li, N. Kanazawa, X. Z. Yu, A. Tsukazaki, M. Kawasaki, M. [51] S. K. Kim, K. Nakata, D. Loss, and Y. Tserkovnyak, Phys. Ichikawa, X. F. Jin, F. Kagawa, and Y. Tokura, Phys. Rev. Lett. Rev. Lett. 122, 057204 (2019). 110, 117202 (2013). [52] S. A. Díaz, T. Hirosawa, J. Klinovaja, and D. Loss, Phys. Rev. [25] K. A. van Hoogdalem, Y. Tserkovnyak, and D. Loss, Phys. Research 2, 013231 (2020). Rev. B 87, 024402 (2013). [53] X. Xing, Y. Zhou, and H. B. Braun, Phys. Rev. Appl. 13, [26] M. Mochizuki, X. Z. Yu, S. Seki, N. Kanazawa, W. Koshibae, 034051 (2020). J. Zang, M. Mostovoy, Y. Tokura, and N. Nagaosa, Nat. Mater. [54] L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev. B 87, 13, 241 (2014). 144101 (2013). [27] A. Mook, B. Göbel, J. Henk, and I. Mertig, Phys.Rev.B95, [55] R. Shindou, R. Matsumoto, S. Murakami, and J.-i. Ohe, Phys. 020401(R) (2017). Rev. B 87, 174427 (2013). [28] D.-S. Han, K. Lee, J.-P. Hanke, Y. Mokrousov, K.-W. Kim, W. [56] A. Mook, J. Henk, and I. Mertig, Phys.Rev.B90, 024412 Yoo, Y. L. W. van Hees, T.-W. Kim, R. Lavrijsen, C.-Y. You, (2014). H. J. M. Swagten, M.-H. Jung, and M. Kläui, Nat. Mater. 18, [57] X. S. Wang, H. W. Zhang, and X. R. Wang, Phys. Rev. Appl. 703 (2019). 9, 024029 (2018). [29] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotechnol. [58] S.-Z. Lin and L. N. Bulaevskii, Phys.Rev.B88, 060404(R) 8, 742 (2013). (2013). [30] R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. [59] A. Roldán-Molina, M. J. Santander, A. S. Nunez, and J. Carpentieri, and G. Finocchio, Sci. Rep. 4, 6784 (2014). Fernández-Rossier, Phys. Rev. B 92, 245436 (2015). [31] B. S. Kim, J. Phys.: Condens. 31, 383001 (2019). [60] R. Takashima, H. Ishizuka, and L. Balents, Phys.Rev.B94, [32] X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 9400 (2015). 134415 (2016). [33] S. Luo, M. Song, X. Li, Y. Zhang, J. Hong, X. Yang, X. Zou, [61] D. N. Aristov and P. G. Matveeva, Phys. Rev. B 94, 214425 N. Xu, and L. You, Nano Lett. 18, 1180 (2018). (2016). [34] W. Kang, X. Chen, D. Zhu, S. Li, Y. Huang, Y. Zhang, and W. [62] S. A. Diaz and D. P. Arovas, arXiv:1604.04010. Zhao, in Proceedings of the 2018 Design, Automation and Test [63] C. Psaroudaki, S. Hoffman, J. Klinovaja, and D. Loss, Phys. in Europe Conference Exhibition (DATE) (IEEE, Piscataway, Rev. X 7, 041045 (2017). NJ, 2018), pp. 119–124. [64] B. Douçot, D. L. Kovrizhin, and R. Moessner, Ann. Phys. [35] D. Pinna, G. Bourianoff, and K. Everschor-Sitte, (NY) 399, 239 (2018). arXiv:1811.12623. [65] A. Derras-Chouk, E. M. Chudnovsky, and D. A. Garanin, [36] J. Zázvorka, F. Jakobs, D. Heinze, N. Keil, S. Kromin, Phys. Rev. B 98, 024423 (2018). S. Jaiswal, K. Litzius, G. Jakob, P. Virnau, D. Pinna, K. [66] C. Psaroudaki and D. Loss, Phys.Rev.Lett.120, 237203 Everschor-Sitte, L. Rózsa, A. Donges, U. Nowak, and M. (2018). Kläui, Nat. Nanotechnol. 14, 658 (2019). [67] O. M. Sotnikov, V. V. Mazurenko, J. Colbois, F. Mila, M. I. [37] M. G. Mankalale, Z. Zhao, J. Wang, and S. S. Sapatnekar, Katsnelson, and E. A. Stepanov, arXiv:1811.10823. IEEE Trans. Devices 66, 1990 (2019). [68] H. Ochoa and Y. Tserkovnyak, Int. J. Mod. Phys. B 33, [38] J. Graf, H. Pfeifer, F. Marquardt, and S. Viola Kusminskiy, 1930005 (2019). Phys.Rev.B98, 241406(R) (2018). [69] C. Psaroudaki, P. Aseev, and D. Loss, Phys.Rev.B100, [39] M. J. Martínez-Pérez and D. Zueco, New J. Phys. 21, 115002 134404 (2019). (2019). [70] V. Lohani, C. Hickey, J. Masell, and A. Rosch, Phys. Rev. X [40] S. V. Kusminskiy, Quantum Magnetism, Spin Waves, and Opti- 9, 041063 (2019). cal Cavities, SpringerBriefs in Physics (Springer International, [71] S. M. Vlasov, P. F. Bessarab, I. S. Lobanov, M. N. Potkina, Cham, 2019), pp. 71–87. V. M. Uzdin, and H. Jónsson, New J. Phys. 22, 083013 [41] C. Schütte and M. Garst, Phys.Rev.B90, 094423 (2014). (2020). [42] F. Ma, Y. Zhou, and W. S. Lew, IEEE Trans. Magn. 51,1 [72] C. Psaroudaki and D. Loss, Phys.Rev.Lett.124, 097202 (2015). (2020). [43] F. Ma, Y. Zhou, H. B. Braun, and W. S. Lew, Nano Lett. 15, [73] L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 36, 1168 (1959) [Sov. 4029 (2015). Phys. JETP 9, 830 (1959)]. [44] A. Roldán-Molina, A. S. Nunez, and J. Fernández-Rossier, [74] A. Zawadowski, J. Ruvalds, and J. Solana, Phys. Rev. A 5, 399 New J. Phys. 18, 045015 (2016). (1972).

033491-17 MOOK, KLINOVAJA, AND LOSS PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

[75] E. H. Graf, V. J. Minkiewicz, H. B. Møller, and L. Passell, [105] S. M. Winter, K. Riedl, P. A. Maksimov, A. L. Chernyshev, A. Phys.Rev.A10, 1748 (1974). Honecker, and R. Valentí, Nat. Commun. 8, 1152 (2017). [76] A. J. Smith, R. A. Cowley, A. D. B. Woods, W. G. Stirling, [106] T. Hong, Y. Qiu, M. Matsumoto, D. A. Tennant, K. Coester, and P. Martel, J. Phys. C 10, 543 (1977). K. P. Schmidt, F. F. Awwadi, M. M. Turnbull, H. Agrawal, and [77] R. Baumgartner, M. Engelhardt, and K. F. Renk, Phys. Rev. A. L. Chernyshev, Nat. Commun. 8, 15148 (2017). Lett. 47, 1403 (1981). [107] P. A. McClarty, X.-Y. Dong, M. Gohlke, J. G. Rau, F. [78] Scattering in Condensed Matter,editedbyW. Pollmann, R. Moessner, and K. Penc, Phys. Rev. B 98, Eisenmenger, K. Laßmann, and S. Döttinger (Springer, Berlin, 060404(R) (2018). 1984). [108] L. Chaix, E. W. Huang, S. Gerber, X. Lu, C. Jia, Y. Huang, [79] H. R. Glyde, M. R. Gibbs, W. G. Stirling, and M. A. Adams, D. E. McNally, Y. Wang, F. H. Vernay, A. Keren, M. Shi, B. Europhys. Lett. 43, 422 (1998). Moritz, Z.-X. Shen, T. Schmitt, T. P. Devereaux, and W.-S. [80] B. Fåk and J. Bossy, J. Low Temp. Phys. 113, 531 (1998). Lee, Phys.Rev.B97, 155144 (2018). [81] M. E. Zhitomirsky and A. L. Chernyshev, Phys. Rev. Lett. 82, [109] P. A. McClarty and J. G. Rau, Phys. Rev. B 100, 100405(R) 4536 (1999). (2019). [82] M. E. Zhitomirsky, Phys.Rev.B73, 100404(R) (2006). [110] J. G. Rau, R. Moessner, and P. A. McClarty, Phys. Rev. B 100, [83] M. B. Stone, I. A. Zaliznyak, T. Hong, C. L. Broholm, and 104423 (2019). D. H. Reich, Nature (London) 440, 187 (2006). [111] T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940). [84] A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. Lett. 97, [112] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). 207202 (2006). [113] T. Moriya, Phys. Rev. 120, 91 (1960). [85] O. F. Syljuåsen, Phys.Rev.B78, 180413(R) (2008). [114] A. Fert and P. M. Levy, Phys. Rev. Lett. 44, 1538 (1980). [86] A. Kreisel, F. Sauli, N. Hasselmann, and P. Kopietz, Phys. Rev. [115] A. Crépieux and C. Lacroix, J. Magn. Magn. Mater. 182, 341 B 78, 035127 (2008). (1998). [87] A. Lüscher and A. M. Läuchli, Phys.Rev.B79, 195102 [116] J. H. Han, J. Zang, Z. Yang, J.-H. Park, and N. Nagaosa, Phys. (2009). Rev. B 82, 094429 (2010). [88] A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. B 79, [117] S.-Z. Lin, C. D. Batista, and A. Saxena, Phys.Rev.B89, 144416 (2009). 024415 (2014). [89] M. Mourigal, M. E. Zhitomirsky, and A. L. Chernyshev, Phys. [118] O. Petrova and O. Tchernyshyov, Phys.Rev.B84, 214433 Rev. B 82, 144402 (2010). (2011). [90] T. Masuda, S. Kitaoka, S. Takamizawa, N. Metoki, K. Kaneko, [119] Dynamics of Solids and Liquids by Neutron Scattering, edited K. C. Rule, K. Kiefer, H. Manaka, and H. Nojiri, Phys. Rev. B by S. W. Lovesey and T. Springer (Springer, Berlin, 1977). 81, 100402(R) (2010). [120] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and Y. Tokura, [91] V. A. Stephanovich and M. E. Zhitomirsky, Europhys. Lett. Phys. Rev. Lett. 109, 037603 (2012). 95, 17007 (2011). [121] Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota, S. Seki, [92] R. L. Doretto and M. Vojta, Phys.Rev.B85, 104416 S. Ishiwata, M. Kawasaki, Y. Onose, and Y. Tokura, Nat. (2012). Commun. 4, 2391 (2013). [93] A. L. Chernyshev, Phys. Rev. B 86, 060401(R) (2012). [122] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, [94] W. T. Fuhrman, M. Mourigal, M. E. Zhitomirsky, and A. L. H. Berger, C. Pfleiderer, and D. Grundler, Nat. Mater. 14, 478 Chernyshev, Phys.Rev.B85, 184405 (2012). (2015). [95] M. E. Zhitomirsky and A. L. Chernyshev, Rev. Mod. Phys. 85, [123] D. Ehlers, I. Stasinopoulos, V. Tsurkan, H.-A. Krug von 219 (2013). Nidda, T. Fehér, A. Leonov, I. Kézsmárki, D. Grundler, and [96] J. Oh, M. D. Le, J. Jeong, J.-h. Lee, H. Woo, W.-Y. Song, T. G. A. Loidl, Phys.Rev.B94, 014406 (2016). Perring, W. J. L. Buyers, S.-W. Cheong, and J.-G. Park, Phys. [124] M. Ikka, A. Takeuchi, and M. Mochizuki, Phys.Rev.B98, Rev. Lett. 111, 257202 (2013). 184428 (2018). [97] V. A. Stephanovich and M. E. Zhitomirsky, Phys. Rev. B 89, [125] R. Chisnell, J. S. Helton, D. E. Freedman, D. K. Singh, R. I. 224415 (2014). Bewley, D. G. Nocera, and Y. S. Lee, Phys.Rev.Lett.115, [98] A. L. Chernyshev, Phys. Rev. B 92, 094409 (2015). 147201 (2015). [99] A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. B 92, [126] L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone, A. I. 144415 (2015). Kolesnikov, Q. Huang, and P. Dai, Phys.Rev.X8, 041028 [100] Z. Z. Du, H. M. Liu, Y. L. Xie, Q. H. Wang, and J.-M. Liu, (2018). Phys.Rev.B92, 214409 (2015). [127] H. Zhang, X. Feng, T. Heitmann, A. I. Kolesnikov, M. B. [101] C. Luo, T. Datta, Z. Huang, and D.-X. Yao, Phys.Rev.B92, Stone, Y.-M. Lu, and X. Ke, Phys.Rev.B101, 100405(R) 035109 (2015). (2020). [102] J. Oh, M. D. Le, H.-H. Nahm, H. Sim, J. Jeong, T. G. Perring, [128] S. Mizukami, D. Watanabe, M. Oogane, Y. Ando, Y. Miura, M. H. Woo, K. Nakajima, S. Ohira-Kawamura, Z. Yamani, Y. Shirai, and T. Miyazaki, J. Appl. Phys. 105, 07D306 (2009). Yoshida, H. Eisaki, S.-W. Cheong, A. L. Chernyshev, and J.-G. [129] S. Mankovsky, D. Ködderitzsch, G. Woltersdorf, and H. Ebert, Park, Nat. Commun. 7, 13146 (2016). Phys. Rev. B 87, 014430 (2013). [103] Z. Z. Du, H. M. Liu, Y. L. Xie, Q. H. Wang, and J.-M. Liu, [130] P. Dürrenfeld, F. Gerhard, J. Chico, R. K. Dumas, M. Phys.Rev.B94, 134416 (2016). Ranjbar, A. Bergman, L. Bergqvist, A. Delin, C. Gould, [104] A. L. Chernyshev and P. A. Maksimov, Phys. Rev. Lett. 117, L. W. Molenkamp, and J. Åkerman, Phys. Rev. B 92, 214424 187203 (2016). (2015).

033491-18 QUANTUM DAMPING OF SKYRMION CRYSTAL … PHYSICAL REVIEW RESEARCH 2, 033491 (2020)

[131] S. Andrieu, A. Neggache, T. Hauet, T. Devolder, A. Hallal, [145] T. Okubo, S. Chung, and H. Kawamura, Phys. Rev. Lett. 108, M. Chshiev, A. M. Bataille, P. Le Fèvre, and F. Bertran, Phys. 017206 (2012). Rev. B 93, 094417 (2016). [146] Y. Kamiya and C. D. Batista, Phys.Rev.X4, 011023 (2014). [132] M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva, H. T. [147] A. O. Leonov and M. Mostovoy, Nat. Commun. 6, 8275 Nembach, O. Eriksson, O. Karis, and J. M. Shaw, Nat. Phys. (2015). 12, 839 (2016). [148] S. von Malottki, B. Dupé, P. F. Bessarab, A. Delin, and S. [133] S. Husain, S. Akansel, A. Kumar, P. Svedlindh, and S. Heinze, Sci. Rep. 7, 12299 (2017). Chaudhary, Sci. Rep. 6, 28692 (2016). [149] S. Hayami and Y. Motome, Phys.Rev.B99, 094420 (2019). [134] A. J. Lee, J. T. Brangham, Y. Cheng, S. P. White, W. T. Ruane, [150] H. D. Rosales, D. C. Cabra, and P. Pujol, Phys.Rev.B92, B. D. Esser, D. W. McComb, P. C. Hammel, and F. Yang, Nat. 214439 (2015). Commun. 8, 234 (2017). [151] M. Li, N. B. Perkins, and I. Rousochatzakis, Phys. Rev. [135] Q. Qin, S. He, W. Song, P. Yang, Q. Wu, Y. P. Feng, and J. Research 1, 013002 (2019). Chen, Appl. Phys. Lett. 110, 112401 (2017). [152] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. [136] J. M. Shaw, E. K. Delczeg-Czirjak, E. R. J. Edwards, Y. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Nat. Kvashnin, D. Thonig, M. A. W. Schoen, M. Pufall, M. L. Phys. 7, 713 (2011). Schneider, T. J. Silva, O. Karis, K. P. Rice, O. Eriksson, and [153] R. F. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. Ellis, H. T. Nembach, Phys.Rev.B97, 094420 (2018). andR.W.Chantrell,J. Phys.: Condens. Matter 26, 103202 [137] L. Bainsla, R. Yilgin, M. Tsujikawa, K. Suzuki, M. Shirai, and (2014). S. Mizukami, J. Phys. D 51, 495001 (2018). [154] J. Colpa, Physica A 93, 327 (1978). [138] I. Stasinopoulos, S. Weichselbaumer, A. Bauer, J. Waizner, H. [155] J. G. Rau, P. A. McClarty, and R. Moessner, Phys.Rev.Lett. Berger, S. Maendl, M. Garst, C. Pfleiderer, and D. Grundler, 121, 237201 (2018). Appl. Phys. Lett. 111, 032408 (2017). [156] E. Rastelli, Statistical Mechanics of Magnetic Excitations [139] M. Halder, A. Chacon, A. Bauer, W. Simeth, S. Mühlbauer, (World Scientific, Singapore, 2011). H. Berger, L. Heinen, M. Garst, A. Rosch, and C. Pfleiderer, [157] R. N. Costa Filho, M. G. Cottam, and G. A. Farias, Phys. Rev. Phys.Rev.B98, 144429 (2018). B 62, 6545 (2000). [140] Q. Tong, F. Liu, J. Xiao, and W. Yao, Nano Lett. 18, 7194 [158] K. Schultheiss, R. Verba, F. Wehrmann, K. Wagner, L. Körber, (2018). T. Hula, T. Hache, A. Kákay, A. A. Awad, V. Tiberkevich, [141] B. Ding, Z. Li, G. Xu, H. Li, Z. Hou, E. Liu, X. Xi, F. Xu, Y. A. N. Slavin, J. Fassbender, and H. Schultheiss, Phys. Rev. Yao, and W. Wang, Nano Lett. 20, 868 (2019). Lett. 122, 097202 (2019). [142] A. K. Behera, S. Chowdhury, and S. R. Das, Appl. Phys. Lett. [159] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 114, 232402 (2019). (1966). [143] J. Liang, W. Wang, H. Du, A. Hallal, K. Garcia, M. Chshiev, [160] F. Bloch, Z. Naturforsch. A 61, 206 (1930). A. Fert, and H. Yang, Phys.Rev.B101, 184401 (2020). [161] S. Seki, M. Garst, J. Waizner, R. Takagi, N. D. Khanh, Y. [144] J. Yuan, Y. Yang, Y. Cai, Y. Wu, Y. Chen, X. Yan, and L. Shen, Okamura, K. Kondou, F. Kagawa, Y. Otani, and Y. Tokura, Phys.Rev.B101, 094420 (2020). Nat. Commun. 11, 256 (2020).

033491-19