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 constants, 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

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2 a r a SUð Þ case described as Eq. (1.2), the term linear in a be expressed as a linear combination jZij ¼ Z ð jÞjx ij background field can be viewed as a Lifshitz invariant term. where rj stands for the position of the site j, and Z ¼ In Sec. III, we study exact skyrmionic solutions of the ðZ1;Z2;Z3ÞT is a complex vector of unit length [31,36]. SUð3Þ gauged CP2 NLσ model in the presence of a Since the state jZij satisfies an over-completeness relation, potential term for the BPS point and solvable line using one can obtain the classical Hamiltonian using the state the BPS arguments. The numerical construction of baby ⊗ ⊗ a r a skyrmion solutions off the solvable line is given in Sec. IV. jZi¼ j jZij ¼ j Z ð jÞjx ij: ð2:3Þ Our conclusions are given in Sec. V. Since Z is normalized and has the gauge degrees of freedom 2 corresponding to the overall phase factor multiplication, it II. GAUGED CP NLσ MODEL FROM 5 1 2 takes values in S =S ≈ CP . In terms of the basis (2.2), the A SPIN SYSTEM SUð3Þ spin operators can be defined as To find Lifshitz invariant terms relevant for the CP2 NLσ 2 Tm λ xa xb ;m 1; 2; …; 8; 2:4 model, we begin to derive an SUð3Þ gauged CP NLσ ¼ð mÞabj ih j ¼ ð Þ model, a generalization of Eq. (1.2), from a spin system on where λm is the mth component of the Gell-Mann matrices. a square lattice. By analogy with Eq. (1.2), the Lifshitz One can check that they satisfy the SUð3Þ commutation invariant, in that case, can be introduced as a term linear in a relation (1.5). The expectation values of the SUð3Þ oper- nondynamical background gauge potential of the gauged ators in the state (2.3) are given by CP2 model. σ m ≡ m r λ ¯ a r b r Following the procedure to obtain a gauged NL model hTj i n ð jÞ¼ð mÞabZ ð jÞZ ð jÞ; ð2:5Þ from a spin system, as discussed in Ref. [19], we consider a ¯ a a generalization of the SU(3) Heisenberg model defined by where Z denotes the complex conjugation of Z . In the m the Hamiltonian context of QCD, the field n is usually termed a color (direction) field [37]. The color field satisfies the J X H − m ˆ n constraints ¼ 2 Ti ðUijÞmnTj ; ð2:1Þ hiji 4 3 m m m p q n n ¼ 3 ;n¼ 2 dmpqn n ; ð2:6Þ where Uˆ is a background field which can be recognized as ij 1 λ λ λ a Wilson line operator along with the link from the point i where dmpq ¼ 4 Trð mf p; qgÞ. Consequently, the number to the point j, which is an element of the SUð3Þ group in the of degrees of freedom of the color field reduces to four. adjoint representation. As in the SUð2Þ case [19], the field Note that, combining the constraints (2.6), one can get the m p q ˆ Casimir identity dmpqn n n ¼ 8=9. Uij may describe effects originated from spin (nematic)- orbital coupling, complicated crystalline structure, and so In terms of the color field, the classical Hamiltonian is on. This Hamiltonian can be viewed as the exchange given by interaction term for the tilted operator T˜ m W TmW−1, J X i ¼ i i i ≡ H − l r ˆ m r m −1 H hZj jZi¼ n ð iÞðUijÞlmn ð jÞ: ð2:7Þ where Wi ∈ SUð3Þ, because one can write WjT W ¼ 2 j j hiji n 3 ðRjÞmnTj where Rj is an element of SUð Þ in the adjoint ˆ T Let us write the position of a site j next to a site i as representation. Clearly, Uij ¼ Ri Rj, where T stands for the rj ¼ ri þ aϵek, where ek is the unit vector in the kth transposition. ϵ 1 Let us now find the classical counterpart of the quantum direction ¼ , and a stands for the lattice constant. ≪ 1 ˆ Hamiltonian (2.1). It can be defined as an expectation value For a , the field Uij can be approximated by the of Eq. (2.1) in a state possessing over completeness, exponential expansion through a path integral representation of the partition ˆ iaϵAmðr Þlˆ U ≈ e k i m function. In order to construct such a state for the spin-1 ij 2 system, it is convenient to introduce the Cartesian basis a 1 iaϵAm r lˆ − Am r An r lˆ lˆ O a3 ; ¼ þ k ð iÞ m 2 k ð iÞ kð iÞ m n þ ð Þ i jx1i¼pffiffiffi ðj þ 1i − j − 1iÞ; 2 ð2:8Þ 1 1 ˆ 2 ffiffiffi 1 − 1 where is the unit matrix and lm are the generators of jx i¼p ðj þ iþj iÞ; 3 ˆ 2 SUð Þ in the adjoint representation, i.e., ðlmÞpq ¼ ifmpq. jx3i¼−ij0i; ð2:2Þ In addition, since the model (2.1) is ferromagnetic, it is natural to assume that nearest-neighbor spins are oriented where jmi¼jS ¼ 1;mi (m ¼ 0, 1). In terms of the in the almost same direction, which allows us to use the Cartesian basis, an arbitrary spin-1 state at a site j can Taylor expansion

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m m m 2 SU n ðrjÞ¼n ðriÞþaϵ∂kn ðriÞþOða Þ: ð2:9Þ III. EXACT SOLUTIONS OF THE ð3Þ GAUGED CP2 NLσ MODEL Replacing theR sum over the lattice sites in Eq. (2.7) by the −2 2 In this section, we derive exact solutions of the integral a d x, we obtain a continuum Hamiltonian, model with the Hamiltonian (2.11) supplemented by a except for a constant term, of the form potential term. We first remark on the validity of the Z variational problem. As discussed in Refs. [20,25] for J 2 ∂ n∂ n − 2 n ∂ n the SUð2Þ case, a surface term, which appears in the H ¼ 8 d x½Trð k k Þ iTrðAk½ ; k Þ process of variation, cannot be ignored if the physical space − n 2 Trð½Ak; Þ; ð2:10Þ is noncompact and the gauge potential Ak does not vanish at the spatial infinity like the DM term. This problem can mλ n mλ 2 where Ak ¼ Ak m and ¼ n m. Similar to its SUð Þ be cured by introducing an appropriate boundary term, counterpart expressed as Eq. (1.2), this Hamiltonian can like [20] SU 3 CP2 also be written as the static energy of an ð Þ gauged Z NLσ model ∓ 4ρ 2 ε ∂ n Z HBoundary ¼ d x jk jTrð AkÞ; ð3:1Þ J 2 n n H ¼ 8 d xTrðDk Dk Þ; ð2:11Þ where ρ ¼ J=8. Here the gauge potential Ak satisfies where Dkn ¼ ∂kn − i½Ak; n is the SUð3Þ covariant derivative. Since the Hamiltonian is given by the SUð3Þ covariant i n∞;A ε n∞; n∞;A 0; 3:2 derivative, Eq. (2.11) is invariant under the SUð3Þ gauge ½ j2 jk½ ½ k ¼ ð Þ transformation where n∞ is the asymptotic value of n at spatial infinity. n → n −1 → −1 ∂ −1 g g ;Ak gAkg þ ig kg ; ð2:12Þ Note that Eq. (3.2) corresponds to the asymptotic form of the BPS equation, which we shall discuss in the next ∈ 3 where g SUð Þ. Note that, however, since the subsection. Hence, all field configurations we consider in Hamiltonian (2.11) does not include kinetic terms for this paper satisfy this equation automatically. the gauge field, like the Yang-Mills term, or the Chern- Since Eq. (3.1) is a surface term, it does not contribute to Simons term, the gauge potential is just a background field, the Euler-Lagrange equation, i.e., the classical Heisenberg not the dynamical one. We suppose that the gauge field is equation. Note that the solutions derived in the following fixed beforehand by the structure of a sample and give the sections satisfy Derrick’s scaling relation with the boundary value by hand, like the SUð2Þ case. The gauge fixing allows term, which is obtained by keeping the background field Ak us to recognize the second term in Eq. (2.10) as a Lifshitz intact under the scaling, i.e., E1 þ 2E0 ¼ 0, where E1 invariant term. denotes the energy contribution from the first derivative We would like to emphasize that we do not deal with 2 terms including the boundary term (3.1) and E0 from no Eq. (2.11) as a gauge theory. Rather, we deem it the CP derivative terms. NLσ model with a Lifshitz invariant, and show the existence of the exact and the numerical solutions. For the baby skyrmion solutions we shall obtain, the color A. BPS solutions field n approaches to a constant value n∞ at spatial infinity Recently, it has been proved that the SUð2Þ gauged so that the physical space R2 can be topologically com- CP1 NLσ model (1.2) possesses BPS solutions in the pactified to S2. Therefore, they are characterized by the presence of a particular potential term [20,24]. Here, we topological degree of the map n∶R2 ∼ S2 ↦ CP2 given by show that BPS solutions also exist in the SUð3Þ gauged 2 Z CP model with a special choice of the potential term, − i 2 ε n ∂ n ∂ n which is given by Q ¼ 32π d x jkTrð ½ j ; k Þ: ð2:13Þ Z 4ρ 2 n Combining with the assumption that the gauge is fixed, it is Hpot ¼ d xTrð F12Þ; ð3:3Þ reasonable to identify this quantity (2.13) with the topological charge in our model.1 where Fjk ¼ ∂jAk − ∂kAj − i½Aj;Ak. As we shall see in the next subsection, the potential term can possess a 1If one extends the model (2.11) with a dynamical gauge field, the topological charge is defined by the SUð3Þ gauge invariant natural physical interpretation for some background quantity which is directly obtained by replacing the partial gauge field. It follows that the Hamiltonian we study here difference in Eq. (2.13) with the covariant derivative. reads

065008-4 ISOLATED SKYRMIONS IN THE CP2 NONLINEAR … PHYS. REV. D 103, 065008 (2021) Z Z −1∂ 2 2 Aþ ¼ ig þg; ð3:9Þ H ¼ ρ d xTrðDknDknÞ4ρ d xTrðnF12Þ Z where g ∈ SLð3; CÞ. Note that Eq. (3.9) is not necessarily a ∓ 4ρ 2 ε ∂ n d x jk jTrð AkÞ; ð3:4Þ pure gauge. Similarly, Eq. (3.8) with the minus sign on the −1 right-hand side can be solved if A− ¼ ig ∂−g. For the where the double-sign corresponds to that of Eq. (3.1). background field (3.9), one finds that the BPS equa- First, let us show that the lower energy bound of tion (3.8) is equivalent to Eq. (3.4) is given by the topological charge (2.13). The 1 first term in Eq. (3.4) can be written as −1 ∂ n˜ − ½n˜ ; ∂ n˜ ¼0; n˜ ¼ gng ; ð3:10Þ Z þ 2 þ ρ 2 n n d xTrðDk Dk Þ because, under the SLð3; CÞ gauge transformation, the Z n → n˜ n −1 → ˜ ρ i 2 fields are changed as ¼ g g and Aþ Aþ ¼ d2x Tr D nD n Tr n;D n 2 −1 ∂ −1 0 ¼ 2 ð k k Þþ 2 ð½ k Þ gAþg þ ig þg ¼ . In the following, we only consider Z Eq. (3.9) to simplify our discussion. ρ 2 2 i In order to solve the equation (3.10), we introduce a ¼ d xTr D n ε ½n;D n 2 j 2 jk k tractable parametrization of the color field Z iρ d2xε Trðn½D n;D nÞ 2 2 jk j k n − ffiffiffi λ † Z ¼ p U 8U ; ð3:11Þ ρ 3 ≥ i 2 ε n n n 2 d x jkTrð ½Dj ;Dk Þ: ð3:5Þ with U ¼ðY1; Y2; ZÞ ∈ SUð3Þ, where Z is the continuum counterpart of the vector Z in Eq. (2.3) and Y1, Y2 are It follows that the equality is satisfied if vectors forming an orthonormal basis for C3 with Z.Upto the gauge degrees of freedom, the components Yi can be n i ε n n 0 Dj 2 jk½ ;Dk ¼ ; ð3:6Þ written as ¯ 3 ¯ 1 ¯ 2 1 2 2 ¯ 2 3 which reduces to Eq. (3.2) at the spatial infinity. Therefore, ð−Z ; 0; Z ÞT ð−Z Z ; 1 − jZ j ; −Z Z ÞT Y1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Y2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : one obtains the lower bound of the form 1 − jZ2j2 1 − jZ2j2 Z ρ ð3:12Þ ≥ 2 ε n n n 8 n H 2 d x½i jkTrð ½Dj ;Dk Þ þ Trð F12Þ Therefore, the vector Z fully defines the color field n. − 8ε ∂ TrðnA Þ jkZ j k Accordingly, we can write iρ 2 ¼ d xεjkTrðn½∂jn; ∂knÞ 2 2 −1 n˜ ¼ − pffiffiffi Wλ8W ; ð3:13Þ 3 ¼ ∓16πρQ; ð3:7Þ

with W ¼ gU ¼ðW1; W2; W3Þ ∈ SLð3; CÞ. It follows that where the corresponding BPS equation is given by Z Eq. (3.6). Note that, unlike the energy bound of the the field , which is the fundamental field of the model, is Z −1W CPN self-dual solutions [7,27], the energy bound (3.7) given by ¼ g 3. Substituting the field (3.13) into the can be negative, and it is not proportional to the absolute equation (3.10), one finds that Eq. (3.10) reduces to the value of the topological charge. coupled equation As is often the case in two-dimensional BPS equations −1 W1 ∂ W3 ¼ 0 [7,20], solutions can be best described in terms of the þ ð3:14Þ 1 2 W−1∂ W 0 complex coordinates z ¼ x ix . Further, we make use 2 þ 3 ¼ of the associated differential operator and background field ∂ 1 ∂ ∓ ∂ 1 ∓ W−1 Y† −1 1 defined as ¼ 2 ð 1 i 2Þ and A ¼ 2 ðA1 iA2Þ. with l ¼ l g (l ¼ , 2). Since the three vectors Then, the BPS equation (3.6) can be written as fY1; Y2; Zg form an orthonormal basis, Eq. (3.14) implies ∂ W βW β þ 3 ¼ 3 where the function is given by 1 β βW−1W W−1∂ W n − n n 0 ¼ 3 3 ¼ 3 þ 3. Therefore, the Eq. (3.10) is D ½ ;D ¼ : ð3:8Þ 2 solved by any configuration satisfying Similar to the SUð2Þ case [20], Eq. (3.8) with a plus sign D W 0 can be solved if the background field has the form þ 3 ¼ ; ð3:15Þ

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D Φ ∂ Φ − Φ−1∂ Φ Φ where þ ¼ þ ð þ Þ for arbitrary nonzero vector Φ. Moreover, we write qffiffiffiffiffiffiffiffiffiffiffi 2 W3 ¼ jW3j w; ð3:16Þ where w is a three component unit vector, i.e., jwj2 ¼ w†w ¼ 1. Then, Eq. (3.15) can be reduced to

D w ≡∂ w − w†∂ w w 0 þ μ ð μ Þ ¼ ; ð3:17Þ which is the very BPS equation of the standard CP2 NLσ FIG. 1. Topological charge density of the axial symmetric solution (3.28) with κ ¼ 1. model. Thus, a general solution of Eq. (3.15), up to the gauge degrees of freedom, is given by 0 1 −1 g1a ðzþÞPaðz−Þ P χ XB C T Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B −1 C w P ¼ @ g2 ðz ÞPaðz−Þ A; ¼ P ; ¼ðP1ðz−Þ;P2ðz−Þ;P3ðz−ÞÞ ; ð3:18Þ 2 2 2 a þ j j jP1j þjP2j þjP3j a −1 g3a ðzþÞPaðz−Þ where P has no overall factor, and Pa is a polynomial in z−. ð3:23Þ Therefore, we finally obtain the solution for the Z field −1 −1 where gab is the ða; bÞ component of the inverse matrix g . Z −1W χ −1w χ −1P −1 ¼ g 3 ¼ g ¼ g ; ð3:19Þ Let Na (Kab) be the highest power in Pa (gab ). Note that −1 though gab are formally represented as power series in zþ, χ where is a normalization factor. the integers Kba are not always infinite; especially, if a positive integer power of Aþ is zero, all of Kba become −1 B. Properties of the BPS solutions finite because g reduces to a polynomial of finite degree θ in zþ. Using the plane polar coordinates fr; g, one can As the BPS bound (3.7) indicates, the lowest energy −1 N þK write g ðz ÞP ðz−Þ ∼ r a ba exp½−iðN − K Þθ at the solution among Eq. (3.19) with a given background ba þ a a ba spatial boundary and find that only the components of the function g possesses the highest topological charge. In highest power in r contribute to the integral (3.22). Since terms of the explicit calculation of the topological charge, we are interested in constructing topological solitons, we we discuss the conditions for the lowest energy solutions. consider the case when the physical space R2 can be The topological charge (2.13) can be written in terms 2 Z of Z as compactified to the sphere S , i.e., the field takes some fixed value on the spatial boundary. Such a compactifica- Z i tion is possible if there is only one pair fNa;Kbag giving − 2 εij D Z †D Z Q ¼ 2π d x ð i Þ j : ð3:20Þ the largest sum Na þ Kba or any pairs fNa;Kbag, sharing the largest sum, have the same value of the difference. For such configurations, the topological charge is given by We employ the constant background gauge field Aþ for simplicity. Then, the matrix g in Eq. (3.9) becomes Q ¼ −Na þ Kba; ð3:24Þ

− where the combination fNa;Kbag yields the largest sum g ¼ exp ð iAþzþÞ; ð3:21Þ among any pairs fNc;Kdcg. This equation (3.24) indicates that the highest topological charge configuration is given so that the components of g−1 are given by power series in by the choice Na ¼ 0 for a particular value of a which gives zþ. It allows us to write Eq. (3.20) as a line integral along the biggest Kba. the circle at spatial infinity We are looking for the lowest energy solutions with an Z explicit background field. As a particular example, let us 1 Q ¼ C; ð3:22Þ consider 2π 1 S∞ A1 ¼ κðλ1 þ λ4 þ λ5Þ;A2 ¼ κðλ2 þ λ4 − λ5Þ; ð3:25Þ with C ¼ −iZ†dZ [27,38], since the one-form C becomes globally well defined. To evaluate the integral in Eq. (3.22), where κ is a constant. Clearly, this choice yields the we write explicitly potential term

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FIG. 2. The expectation values hSai for the solution (3.28) with κ ¼ 1. ffiffiffi p 2 8 2 3 2 1 2 3 4 5 6 7 8 V ¼ 4TrðnF12Þ¼−16 3κ n ¼ 16κ ð2 − 3hðS Þ iÞ; ðn∞;n∞;n∞;n∞;n∞;n∞;n∞;n∞Þ pffiffiffi ð3:26Þ ¼ð0; 0; −1; 0; 0; 0; 0; 1= 3Þ: ð3:29Þ which can be interpreted as an easy-axis anisotropy, or It indicates that n takes the vacuum value in the Cartan quadratic Zeeman term, which naturally appears in con- subalgebra of SUð3Þ. Hence, the vacuum of the model 1 2 3 densed matter physics. In this case, the solution (3.19) can corresponds to a spin nematic, i.e., hS i¼hS i¼hS i¼0 2 2 1 2 3 2 be written as and hðS Þ i¼0; hðS Þ i¼hðS Þ i¼1. Unlike the pure CP2 model, there is no degeneracy between the spin 0 ffiffiffi 1 p πi nematic state and ferromagnetic state in our model because P1 z− 2κz e 4 P3 z− ð Þþ þ ð Þ 3 χ B 2 2 C the SUð Þ global symmetry is broken. As shown in Fig. 2, κ 3πi ffiffiffiffi B zþ C Z ¼ p P2 z− iκz P1 z− pffiffi e 4 P3 z− : Δ @ ð Þþ þ ð Þþ 2 ð Þ A the spin nematic state is partially broken around the soliton because the expectation values hSai become finite. Figure 3 P3ðz−Þ shows that hðSaÞ2i of the solution (3.28) are axially ð3:27Þ symmetric, although the expectation values hSai have angular dependence. Therefore, the solution with the highest topological charge α α α α ∈ C is given by P1 ¼ 1, P2 ¼ 2z− þ 3 with i , and P3 C. Exact solutions off the BPS point being a nonzero constant. Choosing P1 P2 0, one can ¼ ¼ 2 obtain the axially symmetric solution Note that the Hamiltonian (1.1) with B ¼ A admits closed-form analytical solutions [40]. Further, the CP1 BPS 0 ffiffiffi 1 p πi truncation corresponds to the restricted choice of the 2κz e 4 2 þ 4 parameters, B ¼ 2A ¼ κ . The relation B ¼ 2A is referred 1 B 2 2 C κ B κ z 3πi C 2 2 2 2 Z ¼ pffiffiffiffi @ pffiffiþ 4 A; Δ ¼ 1 þ 2κ z z− þ z z−; to as the solvable line, whereas the restriction B ¼ 2A ¼ κ Δ 2 e þ 2 þ 1 is called the BPS point [25]. Here we show that similar restrictions occur in our model. For this purpose, we ð3:28Þ consider the generalized Hamiltonian

ν2 μ2 which possesses the topological charge Q ¼ 2. Note that H ¼ HD þ HL þ HBoundary þ Hani þ Hpot; ð3:30Þ this configuration also satisfies the BPS equation of 2 ν μ the pure CP NLσ model [26,27,31]. Figure 1 shows the where and are real coupling constants. Here, HD 2 distribution of the topological charge (3.20) of this solution indicates the CP Dirichlet term, i.e., the first term in the (3.28) with κ ¼ 1. We find that the topological charge right-hand side (r.h.s.) of Eq. (2.10), and HL does the density has a single peak, although higher charge topo- Lifshitz invariant term which is the second term of that. logical solitons with axial symmetry are likely to possess a Explicitly, these and other terms read volcano structure, see, e.g., Ref. [39]. These highest charge Z solutions give the asymptotic values at spatial infinity of the ρ 2 ∂ n∂ n HD ¼ d xTrð k k Þ; ð3:31Þ color field

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FIG. 3. The expectation values hðSaÞ2i for the solution (3.28) with κ ¼ 1.

Z μ2 ¼ ν2, the Hamiltonian (3.30) supports a family of exact −2 ρ 2 n ∂ n HL ¼ i d xTrðAk½ ; k Þ; ð3:32Þ solutions of the form Z Z ν2 ν2 c ð Þ¼exp ½i Aþzþ ; ð3:37Þ 2 2 2 H ¼ −ρ d x½Trð½A ; n Þ − Trð½A ; n∞ Þ; ð3:33Þ ani k k where c is a three-component complex unit vector. Z Since the solution (3.37) is a BPS solution of the pure 2 2 CP model with the positive topological charge Q, one gets H ¼ 4ρ d x½TrðnF12Þ − Trðn∞F12Þ; ð3:34Þ pot Z ν2 16πρ HD½ ð Þ ¼ Q. In addition, the lower bound at 2 the BPS point (3.7) indicates that Hν2 μ2 1½Zðν ¼ 1Þ ¼ where A is a constant background field, as before. Finally, ¼ ¼ k −16πρQ. Combining these bounds, we find that the total the boundary term H is defined by Eq. (3.1) with the Boundary energy of the solution (3.37) is given by negative sign in the r.h.s., the same as before. Note that we also introduced constant terms in Eqs. (3.33) and (3.34) in 2 2 Hν2 μ2 ½Zðν Þ ¼ 16πρ 1 − Q: ð3:38Þ order to guarantee the finiteness of the total energy. Clearly, ¼ ν2 the Hamiltonian (3.30) is reduced to Eq. (3.4) as we ν2 2 set ν2 ¼ μ2 ¼ 1. Since the energy becomes negative if < , we can expect ν2 The existence of exact solutions of the Hamiltonian that for small values of the coupling , the homogeneous (3.30) with ν2 ¼ μ2 can be easily shown if we rescale vacuum state becomes unstable, and then separated 2D skyrmions (or a skyrmion lattice) emerge as a ground state. the space coordinates as x⃗→ r0x⃗, where r0 is a positive constant, while the background gauge field Ak remains intact. By rescaling, the Hamiltonian (3.30) becomes IV. NUMERICAL SOLUTIONS 2 ν2 μ2 H ¼ HD þ r0ðHL þ HBoundaryÞþr0ð Hani þ HpotÞ: A. Axial symmetric solutions ð3:35Þ In this section, we study baby skyrmion solutions of the Hamiltonian (3.30) with various combinations of the ν2 μ2 ν−2 Setting ¼ and choosing the scale parameter r0 ¼ , coupling constants. Apart from the solvable line, no exact one gets solutions could find analytically, and then we have to solve −2 r0¼ν −2 the equations numerically. Here, we restrict ourselves to the H 2 2 ¼ H þ ν ðH þ H þ H þ H Þ: ν ¼μ D L Boundary ani pot case of the background field given by Eq. (3.25). ð3:36Þ For the background field (3.25), by analogy with the case of the single CP1 magnetic skyrmion solution, we can look Notice that since the solutions (3.19) with Pi being 2 for a configuration described by the axially symmetric arbitrary constants are holomorphic maps from S to ansatz CP2, they satisfy not only the variational equations 0 1 δHν2 μ2 1 ¼ 0, but also the equations δH ¼ 0, where δ Φ θ ¼ ¼ D sin FðrÞ cos GðrÞei 1ð Þ denotes the variation with respect to n with preserving B C @ iΦ2ðθÞ A the constraint (2.6). Therefore, the solutions also satisfy Z ¼ sin FðrÞ sin GðrÞe ; ð4:1Þ −2 r0¼ν the equations δH 2 2 0. This implies that, in the limit cos FðrÞ ν ¼μ ¼

065008-8 ISOLATED SKYRMIONS IN THE CP2 NONLINEAR … PHYS. REV. D 103, 065008 (2021) where F and G (Φ1 and Φ2) are real functions of the plane some properties of numerical solutions in the extended polar coordinates r (θ). model (3.30). The exact solution on the solvable line ν2 ¼ μ2 with axial For our numerical study, it is convenient to introduce symmetry can be written in terms of the ansatz with the the energy unit 8ρ and the length unit κ−1, in order functions to scale the coupling constants. Then, the rescaled components of the Hamiltonian with the ansatz (4.1) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi become ν8κ4r4 ν2κr −1 2ν4κ2 2 −1 Z F ¼ tan r þ 2 ;G¼ tan 2 ; 2 02 2 02 π 3π HD ¼ d x F þ sin FG Φ1 ¼ θ þ ; Φ2 ¼ 2θ þ : ð4:2Þ 4 4 2 sin F Φ_ 2 2 Φ_ 2 2 þ 2 f 1cos G þ 2sin Gg r Further, the solution (3.28) is given by Eq. (4.2) 4 ν2 1 − sin F Φ_ 2 Φ_ 2 2 with ¼ . This configuration is a useful reference 2 ð 1cos G þ 2sin GÞ ; ð4:3Þ point in the configuration space as we discuss below r

Z 2 pffiffiffi π 0 Φ_ d x 0 G 1 H ¼ −2 2 cos θ þ − Φ1 r cos GF − sin 2F sin G þ sin 2F cos G L r 4 2 2 1 − 2 2 2 Φ_ 2 Φ_ − θ Φ − Φ 2 0 2 2 Φ_ Φ_ sin Fsin F cos Gðcos G 1 þ sin G 2Þ sin ð þ 1 2Þ rsin FG þ 2 sin F sin Gð 1 þ 2Þ 4 2 _ 2 _ − sin F sin 2Gðcos GΦ1 þ sin GΦ2Þ ; ð4:4Þ

Z 1 2 2 2 2 1 π 2 2 H ¼ d x 16sin Fcos G cos F − pffiffiffi cos 2Φ1 − Φ2 þ sin 2F sin G þ sin F sin G ani 2 2 4 þ sin22Fð1 þ 2sin2GÞþ8ðcos2F − cos2Gsin2FÞ2 þ 4cos22Gsin4F − 4 ; ð4:5Þ Z Z pffiffiffi 2 2 1 − 3 8 6 2 2 Hpot ¼ d xð n Þ¼ d x cos F; ð4:6Þ where the prime 0 and the dot_ stands for the derivatives with respect to the radial coordinate r and angular coordinate θ, respectively. The system of corresponding Euler-Lagrange equations for Φi can be solved algebraically for an arbitrary set of the coupling constants, and the solutions are π 3π Φ θ Φ 2θ π 1 ¼ þ 4 ; 2 ¼ þ 4 þ m ; ð4:7Þ where m is an integer. Without loss of generality, we choose m ¼ 0 by transferring the corresponding multiple windings of the phase Φ2 to the sign of the profile function G. Then, the system of the Euler-Lagrange equations for the profile functions with the phase factor (4.7) reads

δH δH δH δH D þ L þ ν2 ani þ μ2 pot ¼ 0; δF δF δF δF δH δH δH δH D þ L þ ν2 ani þ μ2 pot ¼ 0; ð4:8Þ δG δG δG δG with

δH 1 þ 3sin2G 2sin2F D ¼ 2rF00 þ 2F0 − sin 2F rG02 þ − ð1 þ sin2GÞ2 ; ð4:9Þ δF r r

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FIG. 4. Plot of the profile functions fF; Gg (left) and the topological charge density (right) of numerical solutions for changing the coupling constant ν2 at μ2 ¼ 1.5. The gray line indicates the quantities of the exact solution (4.2) on the solvable line.

δH pffiffiffi L ¼ −2 2 2sin2Ff−r sin GG0 þ cos G þ cos Gð1 þ sin2GÞð4cos2F − 1Þg δF 3 − 2 0 − 2 2 4 3 2 1 2 r sin FG 2 sin F sin G þ cos Fsin F sin Gð þ sin GÞ ; ð4:10Þ

δH pffiffiffi ani ¼ 2r½4 2 sin Gcos2Gsin2Fð3 − 4sin2FÞ − 4 cos Fsin3Fcos22G δF þ 4 sin 2Ffcos2F − sin2Fcos2Gð1 þ sin2GÞg − sin 2F cos 2Fð1 þ 2sin2GÞ; ð4:11Þ

δH pot ¼ 6r sin 2F; ð4:12Þ δF δH sin2F sin 2G D ¼ 2r sin F2G00 þ 2r sin 2FF0G0 þ 2sin2FG0 − f3 − 2sin2Fð1 þ sin2GÞg ; ð4:13Þ δG r

δH pffiffiffi L ¼ −2½ 2sin2F sin Gf2rF0 þ sin 2Fð1 − 3sin2GÞg δG þ r sin 2FF0 þ sin2Fð1 − 3 cos 2GÞþsin4Fð1 þ 3 cos 2G − 2cos22GÞ; ð4:14Þ

δH pffiffiffi ani ¼ r½8 2 cos Fsin3F cos Gð1 − 3sin2GÞþ16sin4Fcos3G sin G − sin22F sin 2G; ð4:15Þ δG

δ where c and c are some constants implicitly depending Hpot F G ¼ 0: ð4:16Þ on the coupling constants of the model. To see the behavior δG of solutions at large r, we shift the profile functions as ν2 ≠ μ2 We solve the equations for numerically with the π π boundary condition − F − G F ¼ 2 ;G¼ 2 : ð4:19Þ Fð0Þ¼Gð0Þ¼0; lim FðrÞ¼ lim GðrÞ¼π=2; ð4:17Þ r→∞ r→∞ Then, one obtains linearized asymptotic equations on the functions F and G of the forms which the exact solution (4.2) satisfies. This vacuum corresponds to the spin nematic state (3.29). F 0 4F pffiffiffi G F 00 þ − þ 2 2 G0 − − 2ðν2 þ 3μ2ÞF ¼ 0; Let us consider the asymptotic behavior of the solutions r r2 r of the equations (4.8). Near the origin, the leading terms in G0 G pffiffiffi 2F the power series expansion are G00 þ − − 2 2 F 0 þ ¼ 0: r r2 r

F ≈ cFr; G ≈ cGr; ð4:18Þ ð4:20Þ

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TABLE I. The Hamiltonian and topological charge for the numerical solutions with μ2 ¼ 1.5 where “Derrick” denotes the value 2 2 2 ðHL þ HBoundaryÞ=ðν Hani þ μ HpotÞ, which is expected to be −2 by the scaling argument. For ν ¼ 1.5, we used the exact solution (4.2) so that the Derrick and topological charge for ν2 ¼ 1.5 are exact values.

2 2 2 ν HHD HL ν Hani μ Hpot HBoundary Derrick Q 0.1 −117.47 13.51 −136.48 125.49 5.67 −125.67 −2.00 2.00 0.3 −34.02 13.41 −53.60 41.37 6.69 −41.89 −1.99 2.00 0.8 −8.46 13.06 −29.37 14.73 8.82 −15.71 −1.91 2.00 1.5 −4.19 12.57 −16.76 1.09 15.66 −16.76 −2 2

Unfortunately, Eqs. (4.20) may not support an analytical higher topological degrees. Thus, without loss of general- solution. However, these equations imply that the ity, we can set μ2 ¼ ν2. asymptotic behavior of the profile functions is similar to Following the approach discussed in Ref. [3], let us that of the functions (4.2), by a replacement ν2κ with consider a superposition of the two exact solutions ðν2 þ 3μ2Þ=4. Indeed, the asymptotic equations (4.20) above. This superposition is no longer a solution of the depend on such a combination of the coupling constants, Euler-Lagrange equation, except for in the limit of infinite and there may exist an exact solution on the solvable line separation, because there is a force acting on the solitons. with the same character of asymptotic decay as the The interaction energy of two solitons can be written as localized soliton solution of the equation (4.8). − 2 To implement a numerical integration of the coupled EintðRÞ¼HspðRÞ Hexact; ð4:22Þ system of ordinary differential equations (4.8), we introduce the normalized compact coordinate X ∈ ð0; 1 via where HspðRÞ is the energy of two BPS solitons separated by some large but finite distance R from each other, and 1 − X H stands for the static energy of a single exact solution. r ¼ : ð4:21Þ exact X Notice that the lower bound of the Hamiltonian (3.30) with μ2 ¼ ν2 is given The integration was performed by the Newton-Raphson −2 2 −2 2000 ν 2 2 1−ν ≥ 2π 1−2ν method with the mesh point NMESH ¼ . H ¼ Hν ¼μ ¼1 þð ÞHD ð ÞQ; ð4:23Þ In Fig. 4, we display some set of numerical solutions for different values of the coupling ν2 at μ2 ¼ 1.5 and their where the equality is enjoyed only by holomorphic sol- topologicalR charge density Q defined through Q ¼ utions. Therefore, we immediately conclude 2π rQdr. The solutions enjoy Derrick’s scaling relation ≥ 2 and possess a good approximated value of the topological HspðRÞ Hexact; ð4:24Þ charge, as shown in Table I. One observes that as the value 2 → ∞ of the coupling ν becomes relatively small, the function G where the equality is satisfied only at the limit R . is delocalizing while the profile function F is approaching It follows that the interaction energy is always positive its vacuum value everywhere in space except for the origin. for finite separation, and the interaction is repulsive. Since 2 This is an indication that any regular nontrivial solution the exact solution has the topological charge Q ¼ ,it does not exist ν2 ¼ 0. implies that there are no isolated soliton solutions with the topological charge Q ≥ 4 in this model. Note that, however, as the BPS solution (3.19) suggests, there can exist soliton B. Asymptotic behavior solutions with an arbitrary negative charge, which are Asymptotic interaction of solitons is related to the topological excited states on top of the homogeneous overlapping of the tails of the profile functions of well- vacuum state. separated single solitons [3]. Bounded multisoliton configurations may exist if there is an attractive force between V. CONCLUSION two isolated solitons. Considering the above-mentioned soliton solutions of In this paper, we have studied two-dimensional sky- the gauged CP2 NLσ model, we have seen that the exact rmions in the CP2 NLσ model with a Lifshitz invariant term solution (4.2) has the same type of asymptotic decay as any which is an SUð3Þ generalization of the DM term. We have solution of the general system (4.8). Therefore, it is enough shown that the SUð3Þ tilted FM Heisenberg model turns out to examine the asymptotic force between the solutions to be an SUð3Þ gauged CP2 NLσ model in which the term on the solvable line (4.2) to understand whether or not linear in a background gauge field can be viewed as a the Hamiltonian (3.30) supports multisoliton solutions of Lifshitz invariant. We have found exact BPS-type solutions

065008-11 AKAGI, AMARI, SAWADO, and SHNIR PHYS. REV. D 103, 065008 (2021) of the gauged CP2 model in the presence of a potential term antiferromagnetic Heisenberg model where soliton or with a specific value of the coupling constant. The least sphaleron solutions can be constructed [41–43]. energy configuration among the BPS solutions has been We restricted our analysis on the case that the additional 2 μ2 discussed. We have reduced the gauged CP model to the potential term Hpot is balanced or dominant against the 2 ν2 ν2 ≤ μ2 (ungauged) CP model with a Lifshitz invariant by choos- anisotropic potential term Hani, i.e., . We expect ing a background gauge field. In the reduced model, we that a classical phase transition occurs outside of the have constructed an exact solution for a special combina- condition, and it causes instability of the solution. At the tion of coupling constants called the solvable line and moment, the phase structure of the model (3.30) is not clear, numerical solutions for a wider range of them. and we will discuss it in our subsequent work. For numerical study, we chose the background field, Moreover, it has been reported that in some limit of a generating a potential term that can be interpreted as three-component Ginzburg-Landau model [44,45], and of a the quadratic Zeeman term or uniaxial anisotropic term. three-component Gross-Pitaevskii model [46,47], their One can also choose a background field generating the vortex solutions can be well described by planar CP2 Zeeman term; if the background field is chosen as A1 ¼ skyrmions. We believe that our result provides a hint to −κλ7 and A2 ¼ κλ5, the associated potential term is introduce a Lifshitz invariant to the models, and that our 3 proportional to hS i. The Euler-Lagrange equation for solutions find applications not only in SUð3Þ spin systems the extended CP2 model with this background field is but also in superconductors and Bose-Einstein condensates not compatible with the axial symmetric ansatz (4.1). described by the extended models, including the Lifshitz Therefore, a two-dimensional full simulation is required invariant. to obtain a solution with this background field. This problem, numerical simulation for nonaxial symmetric ACKNOWLEDGMENTS solutions in the CP2 model with a Lifshitz invariant, is left to future study. In addition, the construction of a CP2 This work was supported by JSPS KAKENHI Grants skyrmion lattice is a challenging problem. The physical No. JP17K14352, No. JP20K14411, and JSPS Grant-in- interpretation of the Lifshitz invariants is also an impor- Aid for Scientific Research on Innovative Areas “Quantum tant future task. The microscopic derivation of the SUð3Þ Liquid Crystals” (KAKENHI Grant No. JP20H05154). tilted Heisenberg model [21] may enable us to understand Y. S. gratefully acknowledges support by the Ministry of the physical interpretation and physical situation where Education of Russian Federation, project FEWF-2020- the Lifshitz invariant appears. Other future work would 0003. Y. Amari would like to thank Tokyo University of be the extension of the present study to the SUð3Þ Science for its kind hospitality.

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