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Isolated Skyrmions in the CP2 Nonlinear Sigma Model with a Dzyaloshinskii-Moriya Type Interaction

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Isolated Skyrmions in the CP2 Nonlinear Sigma Model with a Dzyaloshinskii-Moriya Type Interaction PHYSICAL REVIEW D 103, 065008 (2021) Isolated skyrmions in the CP2 nonlinear sigma model with a Dzyaloshinskii-Moriya type interaction Yutaka Akagi ,1 Yuki Amari ,2,3 Nobuyuki Sawado ,4 and Yakov Shnir 2 1Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan 2BLTP, JINR, Dubna 141980, Moscow Region, Russia 3Department of Mathematical Physics, Toyama Prefectural University, Kurokawa 5180, Imizu, Toyama 939-0398, Japan 4Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan (Received 2 February 2021; accepted 10 February 2021; published 22 March 2021) We study two dimensional soliton solutions in the CP2 nonlinear sigma model with a Dzyaloshinskii- Moriya type interaction. First, we derive such a model as a continuous limit of the SUð3Þ tilted ferromagnetic Heisenberg model on a square lattice. Then, introducing an additional potential term to the derived Hamiltonian, we obtain exact soliton solutions for particular sets of parameters of the model. The vacuum of the exact solution can be interpreted as a spin nematic state. For a wider range of coupling constants, we construct numerical solutions, which possess the same type of asymptotic decay as the exact analytical solution, both decaying into a spin nematic state. DOI: 10.1103/PhysRevD.103.065008 I. INTRODUCTION matter systems with intrinsic and induced chirality [8–12]. These baby skyrmions, often referred to as magnetic In the 1960s, Skyrme introduced a (3 þ 1)-dimensional skyrmions, were experimentally observed in noncentro- Oð4Þ nonlinear (NL) sigma model [1,2], which is now well symmetric or chiral magnets [13–15]. This discovery known as a prototype of a classical field theory that triggered extensive research on skyrmions in magnetic supports topological solitons (See Ref. [3], for example). materials. This direction is a rapidly growing area both Historically, the Skyrme model has been proposed as a low- theoretically and experimentally [16]. energy effective theory of atomic nuclei. In this framework, A typical stabilizing mechanism of magnetic skyrmions the topological charge of the field configuration is iden- is the existence of the Dzyaloshinskii-Moriya (DM) inter- tified with the baryon number. action [17,18], which stems from the spin-orbit coupling. In The Skyrme model, apart from being considered a good fact, the magnetic skyrmions in chiral magnets can be well candidate for the low-energy QCD effective theory, has described by the continuum effective Hamiltonian attracted much attention in various applications, ranging from string theory and cosmology to condensed matter Z physics. One of the most interesting developments here is 2 J ∇m 2 κm ∇ m − 3 related to a planar reduction of the NLσ model, the so- H ¼ d x 2 ð Þ þ · ð × Þ Bm called baby Skyrme model [4–6]. This (2 þ 1)-dimensional simplified theory resembles the basic properties of the þ Afjmj2 þðm3Þ2g ; ð1:1Þ original Skyrme model in many aspects. The baby Skyrme model finds a number of physical realizations in different branches of modern physics. m r 1 2 3 Originally, it was proposed as a modification of the where ð Þ¼ðm ;m ;m Þ is a three component unit Heisenberg model [4,5,7]. Then, it was pointed out that magnetization vector which corresponds to the spin expect- r skyrmion configurations naturally arise in condensed ation value at position . The first term in Eq. (1.1) is the continuum limit of the Heisenberg exchange interaction, i.e., the kinetic term of the Oð3Þ NLσ model, which is often referred to as the Dirichlet term. The second term there Published by the American Physical Society under the terms of is the DM interaction term, the third one is the Zeeman the Creative Commons Attribution 4.0 International license. coupling with an external magnetic field B, and the last, Further distribution of this work must maintain attribution to 2 3 2 the author(s) and the published article’s title, journal citation, symmetry breaking term Afjmj þðm Þ g represents the and DOI. Funded by SCOAP3. uniaxial anisotropy. 2470-0010=2021=103(6)=065008(13) 065008-1 Published by the American Physical Society AKAGI, AMARI, SAWADO, and SHNIR PHYS. REV. D 103, 065008 (2021) X κ2 2 J It is remarkable that in the limiting case A ¼ = J; H ¼ − TmTm; ð1:4Þ 0 2 i j B ¼ , the Hamiltonian (1.1) can be written as the static hiji version of the SUð2Þ gauged Oð3Þ NLσ model [19,20] m 1 … 8 Z where J is a positive constant, and Ti (m ¼ ; ; ) stand J 2 2 for the SUð3Þ spin operators of the fundamental represen- H ¼ d xð∂ m þ A × mÞ ;k¼ 1; 2; ð1:2Þ 2 k k tation at site i satisfying the commutation relation A −κ 0 0 A l m n with a background gauge field 1 ¼ð =J; ; Þ; 2 ¼ ½Ti;Ti ¼iflmnTi : ð1:5Þ ð0; −κ=J; 0Þ. Though the DM term is usually introduced phenomenologically, a mathematical derivation of the Here, the structure constants are given by flmn ¼ A i Hamiltonian (1.2) with arbitrary k has been developed − 2 Trðλl½λm; λnÞ, where λm are the usual Gell-Mann recently [19]; i.e., it has been shown that the Hamiltonian matrices. can be derived mathematically in a continuum limit of the The SUð3Þ FM Heisenberg model may play an important tilted (quantum) Heisenberg model role in diverse physical systems ranging from string theory X [33] to condensed matter, or quantum optical three-level H − W aW−1 W aW−1 systems [34]. It can be derived from a spin-1 bilinear- ¼ J ð iSi i Þð jSj j Þ; ð1:3Þ hiji biquadratic model with a specific choice of coupling con- stants, so-called FM SUð3Þ point; see, e.g., Ref. [35]. The 3 2 where the sum hiji is taken over the nearest-neighbor sites, SUð Þ spin operators can be defined in terms of the SUð Þ a spin operators Sa (a ¼ 1,2,3)as Si denotes the ath component of spin operators at site i and W ∈ 2 i SUð Þ. It was reported that the tilting Heisenberg 0 1 0 1 model can be derived from a Hubbard model at half-filling T7 S1 B C B C in the presence of spin-orbit coupling [21]. Therefore, the @ 5 A @ − 2 A A T ¼ S ; background field k can still be interpreted as an effect of 2 3 the spin-orbit coupling. T S 0 1 0 1 2 2 2 1 There are two advantages of utilizing the expression T3 ðS Þ − ðS Þ (1.2) for the theoretical study of baby skyrmions in the B C B 1 3 2 C B 8 C B pffiffi ½S · S − 3ðS Þ C presence of the so-called Lifshitz invariant, an interaction B T C B 3 C B 1 C B C term that is linear in a derivative of an order parameter B T C ¼ −B S1S2 þ S2S1 C: ð1:6Þ B 4 C B C [22,23], like the DM term. The first advantage of the form @ T A @ S3S1 þ S1S3 A Eq. (1.2) is that one can study a NLσ model with various T6 2 3 3 2 forms of Lifshitz invariants which are mathematically S S þ S S derived by choice of the background field Ak, although 2 a b ε c Lifshitz invariants have, in general, a phenomenological Using the SUð Þ commutation relation ½Si ;Si ¼i abcSi origin corresponding to the crystallographic handedness where εabc denotes the antisymmetric tensor, one can check of a given sample. The second advantage of the model (1.2) that the operators (1.6) satisfy the SUð3Þ commutation is that it allows us to employ several analytical techniques relation (1.5). developed for the gauged NLσ model. It has been recently In the present paper, we study baby skyrmion solutions reported in Ref. [20] that the Hamiltonian (1.2) with a of an extended CP2 NLσ model composed of the CP2 specific choice of the potential term exactly satisfies the Dirichlet term, a DM type interaction term, i.e., the Lifshitz Bogomol’nyi bound, and the corresponding Bogomol’nyi- invariant, and a potential term. The Lifshitz invariant, Prasad-Sommerfield (BPS) equations have exact closed- instead of being introduced ad hoc in the continuum form solutions [20,24,25]. Hamiltonian, can be derived in a mathematically well- Geometrically, the planar skyrmions are very nicely defined way via consideration of a continuum limit of the 1 described in terms of the CP complex field on the SUð3Þ tilted Heisenberg model. Below we will implement compactified domain space S2 [6]. Further, there are this approach in our derivation of the Lifshitz invariant. In various generalizations of this model; for example, two- the extended CP2 NLσ model, we derive exact soliton dimensional CP2 skyrmions have been studied in the solutions for specific combinations of coupling constants pure CP2 NLσ model [26–28] and in the Faddeev- called the BPS point and solvable line. For a broader range Skyrme type model [29,30]. of coupling constants, we construct solitons by solving the Remarkably, the two-dimensional CP2 NLσ model can Euler-Lagrange equation numerically. be obtained as a continuum limit of the SUð3Þ ferromag- The organization of this paper is the following: In the netic (FM) Heisenberg model [31,32] on a square lattice next section, we derive an SUð3Þ gauged CP2 NLσ model defined by the Hamiltonian from the SUð3Þ tilted Heisenberg model. Similar to the 065008-2 ISOLATED SKYRMIONS IN THE CP2 NONLINEAR … PHYS. REV. D 103, 065008 (2021) 2 a r a SUð Þ case described as Eq.
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