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Quantum States of a Skyrmion in a 2D Antiferromagnet

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Quantum States of a Skyrmion in a 2D Antiferromagnet 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 p ! 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 thep 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(' + γ) 2λr sin(' + γ) r2 − λ2 L = r ;Q r ; ; r2 + λ2 r2 + λ2 r2 + λ2 (2) where r and 'r 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 p ! 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.
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