Quantum States of a Skyrmion in a 2D Antiferromagnet

A. Derras-Chouk, E. M. Chudnovsky, and D. A. Garanin Physics Department, Herbert H. Lehman College and Graduate School, The City University of New York, 250 Bedford Park Boulevard West, Bronx, New York 10468-1589, USA (Dated: April 7, 2021) Quantum states of a skyrmion in a 2D antiferromagnetic lattice are obtained by quantizing the scaling parameter of Belavin-Polyakov model. Skyrmion classical collapse due to violation of the translational invariance of the continuous spin-field model by the lattice is replaced in quantum mechanics by transitions between discrete energy levels of the skyrmion. Rates of transitions due to emission of magnons are computed. Ways of detecting quantization of skyrmion states are discussed.

I. INTRODUCTION oping Holstein-Primakoff transformation of a skyrmion texture, quantum spin excitations of the skyrmion have been obtained13,14 and it was shown that quantum fluc- Skyrmions came to material science1 from high-energy tuations tend to stabilize skyrmion textures. A parti- physics where they were introduced to model atomic cle model of a skyrmion quantum liquid emerging from nuclei2–4. They are prospective candidates for topologi- the melting of a skyrmion crystal has been proposed15. cally protected magnetic memory5–7. Topological stabil- Quantum tunneling of a skyrmion under the energy bar- ity of skyrmions arises from discrete homotopy classes rier created by competing interactions has been stud- of mapping of the continuous field on the continuous ied within semiclassical approach based upon Euclidean geometrical space, e.g., mapping of a three-component action for the spin field16,17. Evidence of quantum constant-length spin field onto the 2D space of a magnetic skyrmion states has been obtained by exact diagonal- film. It relies on the translation (to be exact, conformal) ization of the Heisenberg Hamiltonian of a frustrated invariance of the 2D Heisenberg model. As soon as this ferromagnet18. A review of quantum skyrmionics high- invariance is broken by the crystal lattice, skyrmions be- lighting the relation of the problem to Chern-Simons the- come unstable against collapsing8 and must be stabilized ories and quantum Hall effect has been given in Ref. by additional interactions, such as Dzyaloshiskii-Moriya, 19. Recently, quantum computer simulator has been magnetic anisotropy, Zeeman, etc. In a typical experi- utilized to obtain quantum skyrmion states in a lattice ment the size of the skyrmion is controlled by the mag- model with Heisenberg, Dzyaloshinskii-Moriya, and Zee- netic field. Below a certain size the exchange interaction man interactions20. always wins and the skyrmion collapses9. In this paper we take a different approach to the The observed skyrmion textures typically encom- quantization of the skyrmion field. A Belavin-Polyakov pass thousands or spins. Even the smallest nanoscale skyrmion1 is characterized by a scaling parameter λ that skyrmions experimented with are still comprised of hun- can be roughly interpreted as its size. In a continuous dreds of spins. Such skyrmions were imaged by the spin-field exchange model the energy of the skyrmion is 10 Lorentz transmission electron microscopy and are gen- independent of λ. However, in a discrete model with a erally perceived as classical objects. As the skyrmion finite lattice spacing a the energy acquires8 a term pro- becomes smaller, however, one must expect that at some portional to −(a/λ)2. It leads to the collapse of a classi- point quantum mechanics comes into play. This work cal skyrmion to a point, λ → 0. In quantum mechanics is motivated by the observation that classical collapse of the lattice term can be interpreted as a potential well a skyrmion into a point of a crystal lattice is at odds U(λ) inside which the skyrmion must have quantized en- with quantum mechanics. It contradicts the uncertainty ergy levels. In antiferromagnets, inertia associated with principle the same way as the classical collapse of an the dynamics of the Néel vector allows one to introduce electron onto a proton does. The problem at hand is the conjugate momentum associated with the generalized much more difficult, however, than the problem of the coordinate λ, thus making quantization of the problem hydrogen atom. Huge number of spin degrees of freedom straightforward. It is conceptually similar to the quanti- possessed by the skyrmion resembles the problem of the zation of a string loop collapsing under tension21. arXiv:2104.02212v1 [cond-mat.mes-hall] 6 Apr 2021 many-electron atom for which analytical computation of The article is organized as follows. Classical dynamics quantum states is impossible. of the antiferromagnetic skyrmion in a 2D crystal lattice Some aspects of the quantum behavior of skyrmions is discussed in Section II. Quantization of the Hamilto- have been addressed in the past. Quantum motion of nian of the skyrmion is performed in Section III. Eigen- a skyrmion in the pinning potential has been studied, functions and eigenvalues of the skyrmion in the poten- based upon the analogy of the Thiele dynamics with tial well created by the lattice are obtained. Rates of the the motion of a charged particle in the magnetic field11. transitions between quantized states of the skyrmion, ac- Magnon-skyrmion scattering in chiral magnets has been companied by the radiation of a magnon, are computed addressed by deriving Bolgoliubov - de Gennes Hamilto- in Section IV. Possible systems and experiments are dis- nian from the Lagrangian of the spin field12. By devel- cussed in Section V. 2

II. CLASSICAL SKYRMION ON A LATTICE (2) one has8 √ ! π 2 l/ e dλ2 2πJS2a2 We begin with the exchange Hamiltonian of a 2D an- H = E + ~ ln − . 0 2 p 2 2 tiferromagnet in a continuous spin-field approximation, 2Ja λ2 + a2/6 dt 3(λ + a /6) (5) 1  1  Here we have introduced a large-distance cutoff l due to H = JS2 dxdy ∂ L · ∂ L + ∂ L · ∂ L . (1) 0 2 ˆ c2 t t i i the finite size of the 2D system and a small-distance cutoff due to the discreteness of the crystal lattice. The latter Here J > 0 is a constant of the exchange interaction was chosen such as to eliminate the unphysical disconti- between nearest-neighbor spins of length S, L is a nor- nuities at λ → 0 in the denominators of Eq. (5) and pro- malized Néel vector, and c is the√ speed of antiferromag- vide the zero static energy at λ = 0. This choice is sup- netic spin waves that equals 2 2JSa/~ in a square lat- ported by the direct numerical summation for the energy tice. Summation over the repeated index i = x, y is on the lattice, using skyrmion profile in Fig. (2) with dif- assumed. The first term in Eq. (1) can be interpreted ferent λ, see Fig. 1. Excellent fit of the microscopic many- as the kinetic energy responsible for the inertia of the spin result by the potential V = −4πJS2a2/(6λ2 + a2) spin field in antiferromagnets8,22,23. In the low-energy of Eq. (5) allows one to extend it to the region λ < a. domain, strong antiferromagnetic exchange between an- tiparallel sublattices makes the length of the Néel vector nearly constant, L2 = 1. Hamiltonian (1) is equivalent to the σ-model in relativistic field theory2. Within the continuous field theory based upon Hamil- tonian (1) skyrmions are stable due to the conservation of 1 the topological charge Q = 4π dxdy L·(∂xL×∂yL) that takes values Q = 0, ±1, ±2,...´. Skyrmion with Q = ±1 is given by1

2λr cos(ϕ + γ) 2λr sin(ϕ + γ) r2 − λ2  L = r ,Q r , , r2 + λ2 r2 + λ2 r2 + λ2 (2) where r and ϕr are polar coordinates in the 2D plane, γ is an arbitrary chirality angle, and λ is an arbitrary scal- ing parameter. Crucial for our treatment of the quan- tum problem is observation that λ can be both positive and negative. Its sign is related to the chirality of the skyrmion while its modulus can be interpreted as the 2 Figure 1: Dependence of the energy of the Belavin-Polyakov skyrmion size. The energy of the skyrmion, E0 = 4πJS , is independent of γ and λ. skyrmion on its size, computed numerically in a square lattice of spins. Lattice of a finite spacing a breaks the stability of the skyrmion by making its energy depend on λ. This de- The behavior of λ(t) corresponding to the collapse of pendence was worked out in Ref. 8 within a Heisenberg a classical skyrmion of the initial size λ = 15a in a model with two antiferromagnetic sublattices in a square 0 circular 2D space of radius l = 1000a, that follows from lattice. The lattice contribution to the Hamiltonian is the conservation of energy, 2 √ ! a 2 2 2 dx2 ¯l/ e 4  1 1  Hlat = − JS dxdy ∂i L · ∂i L. (3) ln = − , 24 ˆ p 2 2 2 dt x + 1/6 3 x + 1/6 x0 + 1/6 Treating this term as a perturbation and substituting Eq. (6) (2) into Eq. (3) one obtains for the energy of the skyrmion is shown in Fig. 2. It is derived from the Hamiltonian ¯ with λ & a (5) by choosing x = λ/a, t = JSt/~ and the initial state that starts from rest, dx/dt = 0, with x = x0 = λ0/a.  1  a 2 The dependence of the skyrmion lifetime on the system E = 4πJS2 1 − . (4) 6 λ size ¯l = l/a is weak. Temporal behavior of the collapsing skyrmion shown This result can be generalized24 for an arbitrary Q. It is in Fig. 2 was confirmed by the numerical study of the independent of the chirality angle γ. It shows that due full two-sublattice classical antiferromagnetic Heisenberg to a nonzero a the skyrmion would decrease its energy spin model on a square lattice8. That model also cap- by collapsing towards smaller λ. tured the decay of the topological charge to zero at the The dynamics of the collapse is described by the Hamil- final stage of the collapse. Based upon numerical re- tonian H = H0 + Hlat. For the skyrmion given by Eq. sults it was argued that the collapse of the skyrmion 3

15 that depends logarithmically on l and λ. Its typical value S =1/2 for, e.g., J ∼ 1000K and a ∼ 0.3nm is M ∼ 10−28kg, which is about one hundred electron masses. Up to a log factor it coincides with the mass of the antiferromag- 2 10 netic skyrmion, M = E0/c , that can be obtained by 15 substituting Eq. (2) with x replaced by x − vt into Eq. a 10 )/

t (1).

a 5 λ ( )/

t 0

λ ( -5 5 -10 15 1.5 - l= 1000 0 500 1000 1500 S=1/2 J S t /ℏ 1.0

0 0 100 200 300 400 0.5 ) J S t /ℏ x Ψ ( 0.0 Figure 2: Classical collapse of a skyrmion with from the intial -0.5 n size λ0 = 15a in a circular 2D space of radius l = 1000a. 0 The inset shows periodic oscillations of the skyrmion between 1 states of opposite chirality described by Eq. 6. -1.0 2 3

-2 -1 0 1 2 towards lower energies via the reduction of λ was ac- x companied by the radiation of magnons. If this effect was neglected, the skyrmion would collapse and expand 0.0 in a periodic manner, oscillating in the potential well V = −4πJS2a2/(6λ2 + a2) created by the lattice be- -0.5 tween positive and negative λ corresponding to opposite chiralities, see inset in Fig. 2. -1.0 Here we notice that the classical theory of the skyrmion )

x -1.5 collapse comes into conflict with quantum mechanics. ( The contraction of a skyrmion to a point accompanied V 2.0 by its growing radial momentum contradicts the uncer- - tainty principle. Quantum mechanics should suppress -2.5 continuous radiation of magnons by the skyrmion as it suppresses continuous radiation of electromagnetic waves -3.0 by a classical electron falling onto a proton. This must make skyrmions more stable against the collapse. In the -2 -1 0 1 2 correct description the skyrmion must have quantized en- x ergy levels in the lattice potential and probability distri- bution of the skymion size. Its expectation value must be computed as Figure 3: Upper panel: Eigenfunctions of four lowest-energy states for S = 1/2, l = 1000a scaled to x = λ/a. Lower panel: Energy levels of the skyrmion for S = 1/2 in the units of J in  1/2 2 2 λ¯ = phλ2i = dλψ∗(λ)λ2ψ(λ) (7) the potential well V (x) = −4πS /(6x + 1). ˆ The generalized momentum is p = M(dλ/dt), so that based upon the knowledge of the skyrmion wave function up to a constant E0 the Hamiltonian can be written as ψ(λ). p2 2πJS2a2 H = − . (9) 2M(λ) 3(λ2 + a2/6) III. QUANTUM HAMILTONIAN Imposing quantization as

Hamiltonian (5) can be viewed as a Hamiltonian of a λpˆ − pλˆ = i~ (10) particle with a coordinate λ and a mass and writing √ ! π 2 l/ e M(λ) = ~ ln (8) d Ja2 p 2 2 pˆ = −i (11) λ + a /6 ~dλ 4

n 0 1 2 3 4 5 6 7 8 9 10 11 12

λ¯n/a 0.1518 0.2994 0.4663 0.6840 0.9841 1.410 2.029 2.943 4.318 6.428 9.741 15.09 24.03

En/J -2.581 -1.635 -0.9861 -0.5666 -0.3108 -0.1634 -0.08273 -0.04043 -0.01910 -0.008715 -0.003827 -0.001607 -0.0006391

Table I: Energy levels and rms skyrmion sizes for S = 1/2. we obtain 2π~(n + 1/2), which reduces in our case to solving the equation 1 2πJS2a2 H =p ˆ pˆ − (12) ( √ ! )1/2 2M(λ) 3(λ2 + a2) xn ¯ 2 l/ e  ¯ ¯  1 √ dx ln En − V (x) = n+ 2 " ! # 2 2 ˆ p 2 Ja d l/ e d 2πJS a −xn π x + 1/6 2 = − ln−1 − . 2π dλ pλ2 + a2/6 dλ 3(λ2 + a2/6) (13) with x = λ/a, ¯l = l/a, E¯ = E/J, V¯ = V/J = 2 2 ¯ ¯ It is easy to check that this Hamiltonian is Hermitian −4πS /(6x +1), for En = V (xn), and n = 0, 1, 2, .... Al- ¯ for symmetric and antisymmetric wave functions of the lowed values of n are restricted by the condition xn < l. bound states. Classical limit corresponds to S → ∞ This quasiclassical method works surprisingly well for when the potential energy dominates over the kinetic en- small n, see Fig. 4. The error is 1.3% for the ground ergy, or to l → ∞ when the mass of the “particle” be- state and −4.9% for n = 12 as compared to the values comes infinite. In the numerical work presented below obtained by solving Schr¨odinger equation. For large n the we use S = 1/2 and l = 1000. The dependence on l is decrease of En on increasing n is faster than exp(−n). weak while results for other S are qualitatively similar For each of the quantum states one can compute the because choosing a different S only changes the depth of rms value of λ according to Eq. (7) with the wave func- the potential well. tion ψ(λ) shown in the upper panel of Fig. 3. The first ¯ Eigenstates of the Hamiltonian (12) for the discrete twelve λn are listed in Table I together with the cor- energy spectrum at S = 1/2 and l = 1000a are shown in responding energy levels. An interesting observation is Fig. 3. Finite element discretization method with Arnoldi in order. While distances between the energy levels de- algorithm and shooting have been used and compared crease exponentially as one approaches the top of the with each other. The density of energy levels increas- potential well, the distances between the corresponding ing towards the top of the potential well created by the skyrmion rms sizes increase. This can be qualitatively lattice. These states represent symmetric and antisym- understood by showing positions of the energy levels and metric quantum oscillations between opposite chiralities skyrmion rms sizes together with the attractive poten- 2 2 2 2 corresponding to positive and negative λ. tial, V = −4πJS a /(6λ + a ), see Fig. 5. Correlation ¯ E¯n ∼ V (λn) is apparent from the figure. Due to the quantization of skyrmion states the transitions between densely packed energy levels of a collapsing nanoscale skyrmion must occur via sizable jumps towards smaller skyrmion sizes.

IV. TRANSITIONS BETWEEN QUANTUM STATES OF A SKYRMION

The quantum counterpart of the continuous skyrmion collapse in a classical theory are quantum transitions ¯ from higher to lower energy levels with lower λn. They are accompanied by radiation of magnons that corre- spond to quantized linear waves δL(r, t) of L. Since a skyrmion is an exact extremum of the Hamiltonian (1), it does not interact with linear excitations to the first or- der on δL. This is easy to see from the fact that skyrmion solutions are obtained from the variation of the exchange Figure 4: Energy levels computed by solving the Schr¨odinger Hamiltonian under the condition L2 = 1, equation for the eigenstates and obtained with the help of the 1 Bohr-Sommerfeld quantization condition, Eq. (13). δH = − JS2 dxdy ∂ 2L − (L · ∂ 2L)L · δL, (14) 2 ˆ i i Note that energy eigenstates can be alternatively which provides the extremal equation for the skyrmion 2 2 found from the Bohr-Sommerfeld condition: dλ p(λ) = ∂i L = (L · ∂i L)L. ¸ 5

that Eq. (1) for magnons becomes

X † Hm = ~ωqaq,αaq,α. (17) q,α These two branches of antiferromagnetic magnons are 23 quantized waves of Lx and Ly under the assump- tion that the Néel vector at infinity is directed along the z-axis. It is easy to see that they satisfy the con- ventional commutation relation for an antiferromagnet: 1 [Lx,Lz] = 2 Mz → 0, with Mz being the magnetization. The rate of the transition from the state ψm(λ) with zero magnons to the lower energy state ψn(λ) (n < m) and one magnon is given by the Fermi golden rule25, 2π X Γ = hi|Hˆint|jihj|Hˆint|iiδ(Ei − Ej) (18) ~ i6=j 2π X ˆ 2 = |hψm(λ)1q,α|Hint|ψn(λ)0i| δ(~ωq − ∆mn), ~ q,α

where ∆mn = Em − En. Substituting here Hˆint of Eq. (15) one obtains

2 2π  1  X Γ = JS2a2 | dλψ∗ (λ)ψ (λ) × mn 12 ˆ m n ~ q,α

dxdy ∂2L · h1 |∂2δL |0 i+ ˆ x s q,α x q,α m 2 2
2 ∂y Ls · h1q,α|∂y δLq,α|0mi | δ(~ωq − ∆mn). (19) In this expression

s 2 2 ~c 2 −iq·r iq·r †
∂xδLq,α = − 2 eαqx e aq,α + e aq,α 2JS Aωq Figure 5: Relation between energies and rms sizes of quan- (20) tized skyrmion states is determined by the shape of the po- and thus tential due to the lattice. Upper panel: Small skyrmions with s S = 1/2 in a lattice with l = 1000a. Lower panel: Large c2 skyrmions. 2 ~ 2 iq·r h1q,α|∂xδL|0mi = − 2 eαqxe (21) 2JS Aωq

2 Interaction of skyrmions with magnons comes from the and similar for ∂y δL. This gives lattice term given by Eq. (3). Writing L in that term as 2π  1 2 c2 Ls + δL, where Ls is the skyrmion field of Eq. (2), we 2 2 ~ X ∗ Γmn = JS a dλψ (λ)ψn(λ) 12 2JS2A ˆ m obtain ~ q,α 1 ! 2 2 2 2 2 2 2
q2 q2 Hint = − JS a dxdy ∂xLs · ∂xδL + ∂y Ls · ∂y δL . 2 x iq·r 2 y iq·r 12 ˆ × dxdy ∂xLs · eα e + ∂y Ls · eα e (15) ˆ ωq ωq We select the quantized wave approximation for δL: ×δ(~ωq − ∆mn), s 2 which can be represented as X ~c −iq·r iq·r †
δL = eα e aq,α + e aq,α , (16) 2JS2Aω 2 4 2 q,α q πJS a c X 1 Γ = (M + M ) δ( ω − ∆ ), mn 144A ω x y ~ q mn † q q with α = x, y. Here aq,α, aq,α are operators of creation and annihilation of magnons corresponding to quantum with oscillations of the Néel vector, eα are their polarization 2 vectors, A is the area of the xy-space, and ωq = cq. The ∗ Mα ≡ dλψ (λ)ψm(λ)Fα , α = x, y, (22) factor under the square root in Eq. (16) is chosen such ˆ n 6

n\m 0 1 2 3 4 5 6 7 8 0 0 0.970 × 10−3 0 1.36 × 10−3 0 0.659 × 10−3 0 2.00 × 10−4 0 1 0 0 0.284 × 10−3 0 0.389 × 10−4 0 1.60 × 10−4 0 0.439 × 10−4 2 0 0 0 0.471 × 10−4 0 0.603 × 10−4 0 2.25 × 10−5 0 3 0 0 0 0 0.512 × 10−5 0 0.604 × 10−5 0 2.01 × 10−6 4 0 0 0 0 0 0.389 × 10−6 0 0.417 × 10−6 0 5 0 0 0 0 0 0 2.17 × 10−8 0 2.11 × 10−8 6 0 0 0 0 0 0 0 0.941 × 10−9 0 7 0 0 0 0 0 0 0 0 0.328 × 10−10

Table II: Transition rates in units J/~ from state m to state n for S = 1/2 and l = 1000. where where

F ≡ dxdy ∂2L q2 + ∂2L q2
eiq·r F (q) ≡ dλψ∗ (λ)ψ (λ)λ|λ|K (|λ|q) (27) α ˆ x sα x y sα y mn ˆ m n 1 4 4
iq·r 4 4
= − q + q dxdyL e = − q + q L˜ (λ, q) . with K1 being the modified Bessel function. The transi- x y ˆ sα x y sα tion rate Γmn of Eq. (24) becomes

Further simplification requires computation of the 2 4 ∞ 2π JS a 8 4 4
2 Fourier transform of the skyrmion field Γmn = dq dϕqq sin ϕq + cos ϕq 144~ ˆ0 ˆ0 iq·r   L˜sα (λ, q) ≡ dxdyLsα (λ, r) e . (23) 2 2 2 ∆mn ˆ × (4π) sin (ϕq + γ) |Fmn(q)| δ q − (28) ~c In terms of the latter The integral over ϕq equals 19π/32 independently of γ, 2 2 4 2 q4 + q4
yielding the final compact expression for the rate: πJS a c X x y  ˜ ˜  Γmn = Mx + My δ(~ωq−∆mn), 144A ωq 3 2 q 19π JS 4 2 2 Γmn = (qmna) q Fmn(qmn) , (29) 288 mn where ~ with q ≡ ∆ /( c) and F defined by Eq. (27). 2 mn mn ~ mn ˜ ∗ ˜ Computation of the transition rates reduces, therefore, Mα ≡ dλψm(λ)ψn(λ)Lsα (λ, q) , α = x, y. ˆ to the computation of the coefficients Fnm with the wave functions of the stationary states of the skyrmion found ˜ ˜ By symmetry, Mx and My make equal contributions to in Section III. the rate, which simplifies to Since energies of skyrmion eigenstates, En, decrease 2 4 ∞ 2π very fast on increasing n (see Fig. 4) so do distances JS a 8 4 4
2 Γ = dq dϕ q sin ϕ + cos ϕ between the levels ∆mn for large m and n. In this case the mn 144 ˆ ˆ q q q ~ 0 0 smallness of |λ|qmn = |λ|∆mn/(~c) allows one to replace   ˜ ∆mn K1(|λ|q) in Eq. (27) with its asymptotic form 1/(|λ|q), × Mxδ q − . (24) ∗ ~c leading to Fmn = λmn/q with λmn = dλ λψm(λ)ψn(λ) and to ´ ˜ To calculate Mx, we need the in-plane component of 3 2 19π JS 4 2 the skyrmion field, Γmn = (qmna) |qmnλmn| , m, n  1. (30) 288~ L (r, ϕ ) = L (r) sin (ϕ + γ) . (25) sx r ⊥ r This makes the rates of transitions between high excited 2 2 levels proportional to ∆6 and progressively small with From Eq. (2) one has L⊥(r) = 2λr/(λ + r ). Then mn increasing m and n. Numerically obtained transition ∞ 2π rates for S = 1/2 are shown in Table II. The selection rule L˜ (q, ϕ ) = rL (r)dr dϕ sin (ϕ + γ) related to the parity of the skyrmion states is apparent. sx q ˆ ⊥ ˆ r r 0 0

−iqr cos(ϕr −ϕq ) ×e = −4πi sin (ϕq + γ) λ|λ|K1(|λ|q). V. DISCUSSION and We have studied quantum states of antiferromag- 2 2 2 M˜ x ≡ (4π) sin (ϕq + γ) |Fmn(q)| , (26) netic skyrmions by quantizing the scaling parameter of 7

Belavin-Polyakov model. Our results suggest that en- by thermal fluctuations. When many small skyrmions ergies and sizes of nanoscale skyrmions are quantized. are present, quantization of their energy levels must lead Quantum mechanics must also slow down the collapse of to the peaks in the absorption and noise spectra cor- small skyrmions, making them more stable as was pro- responding to the transitions between the levels. If low posed earlier13. The question is whether quantum states temperatures were required for a precision study of quan- of skyrmions and transitions between them can be ob- tization of skyrmion states a practical question would be served in experiment. how to create a sufficient number of antiferromagnetic Evidence of skyrmions has been reported in parental skyrmions at such temperatures. One way to do it could compounds of high-temperature superconductors26. In be via rapid cooling of the sample accompanied by quan- fact, many of these materials can be ideal systems for tum relaxometry28 of the antiferromagnetic state that application of our theory as they consist of weakly in- settles in. teracting 2D antiferromagnetic layers of spins 1/2 in In this article we have not addressed quantum me- a square lattice. The antiferromagnetic superexchange chanics of ferromagnetic skyrmions. In the absence interaction27 of copper spins via oxygen in CuO layers of interactions other than the Heisenberg exchange, of La2CuO4 is 140meV. This places lifetimes of the low- the ferromagnetic dynamics, unlike antiferromagnetic lying excited states of the skyrmion in the picosecond dynamics, is massless. Consequently, the quantization to nanosecond range. Transitions between upper excited method we applied, cannot be easily extended to a states have much lower probability. ferromagnet. This already showed up in our study of It should be emphasized that our model is not catching the classical collapse8. The collapse of the antiferromag- the final stage of the decay of the skyrmion accompanied netic skyrmion was investigated by two methods, via by the disappearance of its topological charge when it dynamical equation for λ(t) and by solving numerically decreases to the atomic size. This requires a different the full two-sublattice Heisenberg spin model with the mechanism. In classical mechanics on the lattice, it must initial state containing a skyrmion. Excellent agreement occur via instability of the skyrmion shape that breaks its between the two methods was achieved. On the contrary, radial symmetry. Quantum mechanics efecively switches the study of the collapse of the ferromagnetic skyrmion the skyrmion model from 2 to 2+1 dimensions where relied on the second method only. The observed topological charge is not conserved but working it out dynamics was very different, with the lifetime of the 2 5 may require more than one degree of freedom. We have ferromagnetic skyrmion scaling as [~/(JS )](λ0/a) 2 2 not attempted to solve this problem here. compared to [~/(JS )](λ0/a) for the antiferromagnetic Quantization of the quasiclassical states of skyrmions skyrmion. Due to its complexity the full spin model, that are large compared to the lattice spacing must be however, is not well suited for computing energy levels of captured by our model with reasonable accuracy. For a small ferromagnetic skyrmion. Finding a method for a such skyrmions the distance between adjacent energy lev- 2D ferromagnet similar to that for a 2D antiferromagnet, els is of order ∆ ∼ |E| ∼ 2πJS2a2/(3λ2). Transition as well as inclusion in the model of other interactions from classical to quantum dynamics should occur at tem- such as Dzyaloshinskii-Moriya, dipole-dipole, Zeeman, peratures satisfying T . ∆, that is, for sizes satisfying magnetic anisotropy, remains a challenging task. p 2 λ/a . 2πJS /(3T ). For La2CuO4 this gives λ . 3a at room temperature and λ . 30a at helium tempera- ture. The latter makes observation of quantum behavior of skyrmions promising at low temperature. If the col- lapse of a skyrmion could somehow be visualized with the VI. ACKNOWLEDGEMENTS help of modern imaging techniques its quantum nature would reveal itself in a large jump from a nanometer size This work has been supported by the Grant No. DE- to the atomic size. FG02-93ER45487 funded by the U.S. Department of En- At elevated temperatures skyrmions would be created ergy, Office of Science.

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