Isolated Skyrmions in the CP^2 nonlinear σ-model with a Dzyaloshinskii-Moriya type interaction
aa r X i v : . [ h e p - t h ] F e b Isolated Skyrmions in the
C P nonlinear σ -model with aDzyaloshinskii-Moriya type interaction Yutaka Akagi , Yuki Amari , Nobuyuki Sawado , and Yakov Shnir Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo,Tokyo 113-0033, Japan BLTP, JINR, Dubna 141980, Moscow Region, Russia Department of Mathematical Physics, Toyama Prefectural University, Kurokawa 5180,Imizu, Toyama, 939-0398, Japan Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
Abstract
We study two dimensional soliton solutions in the CP nonlinear σ -model with a Dzyaloshinskii-Moriyatype interaction. First, we derive such a model as a continuous limit of the SU (3) tilted ferromagneticHeisenberg model on a square lattice. Then, introducing an additional potential term to the derived Hamil-tonian, we obtain exact soliton solutions for particular sets of parameters of the model. The vacuum of theexact solution can be interpreted as a spin nematic state. For a wider range of coupling constants, we con-struct numerical solutions, which possess the same type of asymptotic decay as the exact analytical solution,both decaying into a spin nematic state. In the 1960s, Skyrme introduced a (3+1)-dimensional O (4) nonlinear (NL) σ -model [1, 2] which is now well-known as a prototype of a classical field theory that supports topological solitons (See Ref. [3], for example).Historically, the Skyrme model has been proposed as a low-energy effective theory of atomic nuclei. In thisframework, the topological charge of the field configuration is identified with the baryon number.The Skyrme model, apart from being considered a good candidate for the low-energy QCD effective theory,has attracted much attention in various applications, ranging from string theory and cosmology to condensedmatter physics. One of the most interesting developments here is related to a planar reduction of the NL σ -model, the so-called baby Skyrme model [4–6]. This (2+1)-dimensional simplified theory resembles the basicproperties of the original Skyrme model in many aspects.The baby Skyrme model finds a number of physical realizations in different branches of modern physics.Originally, it was proposed as a modification of the Heisenberg model [4, 5, 7]. Then, it was pointed outthat Skyrmion configurations naturally arise in condensed matter systems with intrinsic and induced chirality[8–12]. These baby Skyrmions, often referred to as magnetic Skyrmions, were experimentally observed innon-centrosymmetric or chiral magnets [13–15]. This discovery triggered extensive research on Skyrmions inmagnetic materials. This direction is a rapidly growing area both theoretically and experimentally [16].A typical stabilizing mechanism of magnetic skyrmions is the existence of Dzyaloshinskii-Moriya (DM) in-teraction [17, 18], which stems from the spin-orbit coupling. In fact, the magnetic Skyrmions in chiral magnetscan be well described by the continuum effective Hamiltonian H = Z d x (cid:20) J ∇ m ) + κ m · ( ∇ × m ) − Bm + A n | m | + (cid:0) m (cid:1) o(cid:21) , (1.1)1here m ( r ) = ( m , m , m ) is a three component unit magnetization vector which corresponds to the spinexpectation value at position r . The first term in Eq. (1.1) is the continuum limit of the Heisenberg exchangeinteraction, i.e., the kinetic term of the O (3) NL σ -model, which is often referred to as the Dirichlet term. Thesecond term there is the DM interaction term, the third one is the Zeeman coupling with an external magneticfield B , and the last, symmetry breaking term A n | m | + ( m ) o represents the uniaxial anisotropy.It is remarkable that in the limiting case A = κ / J, B = 0 , the Hamiltonian (1.1) can be written as thestatic version of the SU (2) gauged O (3) NL σ -model [19, 20] H = J Z d x ( ∂ k m + A k × m ) , k = 1 , (1.2)with a background gauge field A = ( − κ/J, , , A = (0 , − κ/J, . Though the DM term is usuallyintroduced phenomenologically, a mathematical derivation of the Hamiltonian (1.2) with arbitrary A k hasbeen developed recently [19], i.e.; it has been shown that the Hamiltonian can be derived mathematically in acontinuum limit of the tilted (quantum) Heisenberg model H = − J X h ij i (cid:0) W i S ai W − i (cid:1) (cid:0) W j S aj W − j (cid:1) , (1.3)where the sum h ij i is taken over the nearest-neighbor sites, S ai denotes the a -th component of spin operatorsat site i and W i ∈ SU (2) . It was reported that the tilting Heisenberg model can be derived from a Hubbardmodel at half-filling in the presence of spin-orbit coupling [21]. Therefore, the background field A k can stillbe interpreted as an effect of the spin-orbit coupling.There are two advantages of utilizing the expression (1.2) for the theoretical study of baby Skyrmionsin the presence of the so-called Lifshitz invariant, an interaction term which is linear in a derivative of anorder parameter [22, 23], like the DM term. The first advantage of the form Eq. (1.2) is that one can studya NL σ -model with various form of Lifshitz invariants which are mathematically derived by choice of thebackground field A k , although Lifshitz invariants have, in general, a phenomenological origin correspondingto the crystallographic handedness of a given sample. The second advantage of the model (1.2) is that itallows us to employ several analytical techniques developed for the gauged NL σ -model. It has been recentlyreported in Ref. [20] that the Hamiltonian (1.2) with a specific choice of the potential term exactly satisfiesthe Bogomol’nyi bound, and the corresponding Bogomol’nyi-Prasad-Sommerfield (BPS) equations have exactclosed-form solutions [20, 24, 25].Geometrically, the planar Skyrmions are very nicely described in terms of the CP complex field on thecompactified domain space S [6]. Further, there are various generalizations of this model; for example,two-dimensional CP Skyrmions have been studied in the pure CP NL σ -model [26–28] and in the Faddeev-Skyrme type model [29, 30].Remarkably, the two dimensional CP NL σ -model can be obtained as a continuum limit of the SU (3) ferromagnetic (FM) Heisenberg model [31, 32] on a square lattice defined by the Hamiltonian H = − J X h ij i T mi T mj , (1.4)where J is a positive constant, and T mi ( m = 1 , ..., ) stand for the SU (3) spin operators of the fundamentalrepresentation at site i satisfying the commutation relation (cid:2) T li , T mi (cid:3) = if lmn T ni . (1.5)2ere, the structure constants are given by f lmn = − i Tr ( λ l [ λ m , λ n ]) , where λ m are the usual Gell-Mannmatrices.The SU (3) FM Heisenberg model may play an important role in diverse physical systems ranging fromstring theory [33] to condensed matter, or quantum optical three-level systems [34]. It can be derived from aspin-1 bilinear-biquadratic model with a specific choice of coupling constants, so-called FM SU (3) point, see,e.g., Ref. [35]. The SU (3) spin operators can be defined in terms of the SU (2) spin operators S a ( a = 1 , , )as T T T = S − S S , T T T T T = − ( S ) − ( S ) √ h S · S − S ) i S S + S S S S + S S S S + S S . (1.6)Using the SU (2) commutation relation (cid:2) S ai , S bi (cid:3) = iε abc S ci where ε abc denotes the anti-symmetric tensor, onecan check that the operators (1.6) satisfy the SU (3) commutation relation (1.5).In the present paper, we study baby Skyrmion solutions of an extended CP NL σ -model composed of the CP Dirichlet term, a DM type interaction term, i.e., the Lifshitz invariant, and a potential term. The Lifshitzinvariant, instead of being introduced ad hoc in the continuum Hamiltonian, can be derived in a mathematicallywell-defined way via consideration of a continuum limit of the SU (3) tilted Heisenberg model. Below wewill implement this approach in our derivation of the Lifshitz invariant. In the extended CP NL σ -model, wederive exact soliton solutions for specific combinations of coupling constants called the BPS point and solvableline. For a broader range of coupling constants, we construct solitons by solving the Euler-Lagrange equationnumerically.The organization of this paper is the following: In the next section, we derive an SU (3) gauged CP NL σ -model from the SU (3) tilted Heisenberg model. Similar to the SU (2) case described as Eq. (1.2), the termlinear in a background field can be viewed as a Lifshitz invariant term. In Sec. 3, we study exact Skyrmionicsolutions of the SU (3) gauged CP NL σ -model in the presence of a potential term for the BPS point andsolvable line using the BPS arguments. The numerical construction of baby Skyrmion solutions off the solvableline is given in Sec. 4. Our conclusions are given in Sec. 5. C P NL σ -model from a spin system To find Lifshitz invariant terms relevant for the CP NL σ -model, we begin to derive an SU (3) gauged CP NL σ -model, a generalization of Eq. (1.2), from a spin system on a square lattice. By analogy with Eq. (1.2),the Lifshitz invariant, in that case, can be introduced as a term linear in a non-dynamical background gaugepotential of the gauged CP model.Following the procedure to obtain a gauged NL σ -model from a spin system, as discussed in Ref. [19], weconsider a generalization of the SU (3) Heisenberg model defined by the Hamiltonian H = − J X h ij i T mi ( ˆ U ij ) mn T nj , (2.1)where ˆ U ij is a background field which can be recognized as a Wilson line operator along with the link fromthe point i to the point j , which is an element of the SU (3) group in the adjoint representation. As in the SU (2) case [19], the field ˆ U ij may describe effects originated from spin (nematic)-orbital coupling, complicatedcrystalline structure, and so on. This Hamiltonian can be viewed as the exchange interaction term for the tilted3perator ˜ T mi = W i T mi W − i where W i ∈ SU (3) , because one can write W j T mj W − j = ( R j ) mn T nj where R j isan element of SU (3) in the adjoint representation. Clearly, ˆ U ij = R T i R j , where T stands for the transposition.Let us now find the classical counterpart of the quantum Hamiltonian (2.1). It can be defined as an ex-pectation value of Eq. (2.1) in a state possessing over-completeness, through a path integral representation ofthe partition function. In order to construct such a state for the spin-1 system, it is convenient to introduce theCartesian basis (cid:12)(cid:12) x (cid:11) = i √ | +1 i − |− i ) , (cid:12)(cid:12) x (cid:11) = 1 √ | +1 i + |− i ) , (cid:12)(cid:12) x (cid:11) = − i | i , (2.2)where | m i = | S = 1 , m i ( m = 0 , ± ). In terms of the Cartesian basis, an arbitrary spin-1 state at a site j can be expressed as a linear combination | Z i j = Z a ( r j ) | x a i j where r j stands for the position of the site j ,and Z = ( Z , Z , Z ) T is a complex vector of unit length [31, 36]. Since the state | Z i j satisfies an over-completeness relation, one can obtain the classical Hamiltonian using the state | Z i = ⊗ j | Z i j = ⊗ j Z a ( r j ) | x a i j . (2.3)Since Z is normalized and has the gauge degrees of freedom corresponding to the overall phase factor multi-plication, it takes values in S /S ≈ CP . In terms of the basis (2.2), the SU (3) spin operators can be definedas T m = ( λ m ) ab | x a i (cid:10) x b (cid:12)(cid:12) m = 1 , , · · · , , (2.4)where λ m is the m -th component of the Gell-Mann matrices. One can check that they satisfy the SU (3) commutation relation (1.5). The expectation values of the SU (3) operators in the state (2.3) are given by h T mj i ≡ n m ( r j ) = ( λ m ) ab ¯ Z a ( r j ) Z b ( r j ) , (2.5)where ¯ Z a denotes the complex conjugation of Z a . In the context of QCD, the field n m is usually termed a color(direction) field [37]. The color field satisfies the constraints n m n m = 43 , n m = 32 d mpq n p n q , (2.6)where d mpq = Tr ( λ m { λ p , λ q } ) . Consequently, the number of degrees of freedom of the color field reducesto four. Note that, combining the constraints (2.6), one can get the Casimir identity d mpq n m n p n q = 8 / .In terms of the color field, the classical Hamiltonian is given by H ≡ h Z | H | Z i = − J X h ij i n l ( r i )( ˆ U ij ) lm n m ( r j ) . (2.7)Let us write the position of a site j next to a site i as r j = r i + aǫ e k where e k is the unit vector in the k -thdirection, ǫ = ± , and a stands for the lattice constant. For a ≪ , the field ˆ U ij can be approximated by theexponential expansion ˆ U ij ≈ e iaǫA mk ( r i )ˆ l m = + iaǫA mk ( r i )ˆ l m − a A mk ( r i ) A nk ( r i )ˆ l m ˆ l n + O ( a ) , (2.8)where is the unit matrix and ˆ l m are the generators of SU (3) in the adjoint representation, i.e., (ˆ l m ) pq = if mpq .In addition, since the model (2.1) is ferromagnetic, it is natural to assume that nearest-neighbor spins areoriented in the almost same direction, which allows us to use the Taylor expansion n m ( r j ) = n m ( r i ) + aǫ∂ k n m ( r i ) + O ( a ) . (2.9)4eplacing the sum over the lattice sites in Eq. (2.7) by the integral a − Z d x , we obtain a continuum Hamil-tonian, except for a constant term, of the form H = J Z d x (cid:2) Tr ( ∂ k n ∂ k n ) − i Tr ( A k [ n , ∂ k n ]) − Tr (cid:0) [ A k , n ] (cid:1)(cid:3) , (2.10)where A k = A mk λ m and n = n m λ m . Similar to its SU (2) counterpart expressed as Eq. (1.2), this Hamiltoniancan also be written as the static energy of an SU (3) gauged CP NL σ -model H = J Z d x Tr ( D k n D k n ) , (2.11)where D k n = ∂ k n − i [ A k , n ] is the SU (3) covariant derivative. Since the Hamiltonian is given by the SU (3) covariant derivative, Eq. (2.11) is invariant under the SU (3) gauge transformation n → g n g − , A k → gA k g − + ig∂ k g − , (2.12)where g ∈ SU (3) . Note that, however, since the Hamiltonian (2.11) does not include kinetic terms for thegauge field, like the Yang-Mills term, or the Chern-Simons term, the gauge potential is just a background field,not the dynamical one. We suppose that the gauge field is fixed beforehand by the structure of a sample and givethe value by hand, like the SU (2) case. The gauge fixing allows us to recognize the second term in Eq. (2.10)as a Lifshitz invariant term.We would like to emphasize that we do not deal with Eq. (2.11) as a gauge theory. Rather, we deem it the CP NL σ -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 field n approaches to a constant value n ∞ at spatialinfinity so that the physical space R can be topologically compactified to S . Therefore, they are characterizedby the topological degree of the map n : R ∼ S CP given by Q = − i π Z d x ε jk Tr ( n [ ∂ j n , ∂ k n ]) . (2.13)Combining with the assumption that the gauge is fixed, it is reasonable to identify this quantity (2.13) with thetopological charge in our model . SU (3) gauged C P NL σ -model In this section, we derive exact solutions of the model with the Hamiltonian (2.11) supplemented by a potentialterm. We first remark on the validity of the variational problem. As discussed in Refs. [20, 25] for the SU (2) case, a surface term, which appears in the process of variation, cannot be ignored if the physical space is non-compact and the gauge potential A k does not vanish at the spatial infinity like the DM term. This problem canbe cured by introducing an appropriate boundary term, like [20] H Boundary = ∓ ρ Z d x ε jk ∂ j Tr( n A k ) , (3.1)where ρ = J/ . Here the gauge potential A k satisfies [ n ∞ , A j ] ± i ε ij [ n ∞ , [ n ∞ , A k ]] = 0 , (3.2) If one extends the model (2.11) with a dynamical gauge field, the topological charge is defined by the SU (3) gauge invariantquantity which is directly obtained by replacing the partial difference in Eq. (2.13) with the covariant derivative. n ∞ is the asymptotic value of n at spatial infinity. Note that Eq. (3.2) corresponds to the asymptotic formof the BPS equation, which we shall discuss in the next subsection. Hence, all field configurations we considerin this paper satisfy this equation automatically.Since (3.1) is a surface term, it does not contribute to the Euler-Lagrange equation, i.e., the classical Heisen-berg equation. Note that the solutions derived in the following sections satisfy Derrick’s scaling relation withthe boundary term, which is obtained by keeping the background field A k intact under the scaling, i. e., E + 2 E = 0 where E denotes the energy contribution from the first derivative terms including the boundaryterm (3.1) and E from no derivative terms. Recently, it has been proved that the SU (2) gauged CP NL σ -model (1.2) possesses BPS solutions in thepresence of a particular potential term [20, 24]. Here, we show that BPS solutions also exist in the SU (3) gauged CP model with a special choice of the potential term, which is given by H pot = ± ρ Z d x Tr ( n F ) , (3.3)where F jk = ∂ j A k − ∂ k A j − i [ A j , A k ] . As we shall see in the next subsection, the potential term can possess anatural physical interpretation for some background gauge field. It follows that the Hamiltonian we study herereads H = ρ Z d x Tr ( D k n D k n ) ± ρ Z d x Tr ( n F ) ∓ ρ Z d x ε jk ∂ j Tr( n A k ) , (3.4)where the double-sign corresponds to that of Eq. (3.1).First, let us show that the lower energy bound of Eq. (3.4) is given by the topological charge (2.13). Thefirst term in Eq. (3.4) can be written as ρ Z d x Tr ( D k n D k n ) = ρ Z d x " Tr ( D k n D k n ) + (cid:18) i (cid:19) Tr (cid:0) [ n , D k n ] (cid:1) = ρ Z d x Tr (cid:18) D j n ± i ε jk [ n , D k n ] (cid:19) ± iρ Z d x ε jk Tr ( n [ D j n , D k n ]) ≥ ± iρ Z d x ε jk Tr ( n [ D j n , D k n ]) . (3.5)It follows that the equality is satisfied if D j n ± i ε jk [ n , D k n ] = 0 , (3.6)which reduces to Eq. (3.2) at the spatial infinity. Therefore, one obtains the lower bound of the form H ≥ ± ρ Z d x [ iε jk Tr ( n [ D j n , D k n ]) + 8Tr ( n F ) − ε jk ∂ j Tr ( n A k )]= ± iρ Z d x ε jk Tr ( n [ ∂ j n , ∂ k n ])= ∓ πρ Q, (3.7)where the corresponding BPS equation is given by Eq. (3.6). Note that, unlike the energy bound of the CP N self-dual solutions [7, 27], the energy bound (3.7) can be negative, and it is not proportional to the absolutevalue of the topological charge. 6s is often the case in two-dimensional BPS equations [7, 20], solutions can be best described in termsof the complex coordinates z ± = x ± ix . Further, we make use of the associated differential operator andbackground field defined as ∂ ± = ( ∂ ∓ i∂ ) and A ± = ( A ∓ iA ) . Then, the BPS equation (3.6) can bewritten as D ± n −
12 [ n , D ± n ] = 0 . (3.8)Similar to the SU (2) case [20], Eq. (3.8) with a plus sign can be solved if the background field has the form A + = ig − ∂ + g, (3.9)where g ∈ SL (3 , C ) . Note that Eq. (3.9) is not necessarily a pure gauge. Similarly, Eq. (3.8) with the minussign on the right-hand side can be solved if A − = ig − ∂ − g . For the background field (3.9), one finds that theBPS equation (3.8) is equivalent to ∂ + ˜ n −
12 [˜ n , ∂ + ˜ n ] = 0 , ˜ n = g n g − , (3.10)because, under the SL (3 , C ) gauge transformation, the fields are changed as n → ˜ n = g n g − and A + → ˜ A ± = gA + g − + ig∂ ± g − = 0 . In the following, we only consider Eq. (3.9) to simplify our discussion.In order to solve the equation (3.10), we introduce a tractable parameterization of the color field n = − √ U λ U † , (3.11)with U = ( Y , Y , Z ) ∈ SU (3) , where Z is the continuum counter part of the vector Z in Eq. (2.3) and Y , Y are vectors forming an orthonormal basis for C with Z . Up to the gauge degrees of freedom, the components Y i can be written as Y = (cid:0) − ¯ Z , , ¯ Z (cid:1) T p − | Z | , Y = (cid:0) − ¯ Z Z , − | Z | , − ¯ Z Z (cid:1) T p − | Z | . (3.12)Therefore, the vector Z fully defines the color field n . Accordingly, we can write ˜ n = − √ W λ W − , (3.13)with W = gU = ( W , W , W ) ∈ SL (3 , C ) . It follows that the field Z , which is the fundamental field of themodel, is given by Z = g − W . Substituting the field (3.13) into the equation (3.10), one finds that Eq. (3.10)reduces to the coupled equation ( W − ∂ + W = 0 W − ∂ + W = 0 , (3.14)where W − l = Y † l g − ( l = 1 , . Since the three vectors { Y , Y , Z } form an orthonormal basis, Eq. (3.14)implies ∂ + W = β W where the function β is given by β = β W − W = W − ∂ + W . Therefore, theequation (3.10) is solved by any configuration satisfying D + W = 0 , (3.15)where D + Φ = ∂ + Φ − ( Φ − ∂ + Φ ) Φ for arbitrary non-zero vector Φ . Moreover, we write W = p | W | w , (3.16)7here w is a three component unit vector, i.e. | w | = w † w = 1 . Then, Eq. (3.15) can be reduced to D + w ≡ ∂ µ w − (cid:0) w † ∂ µ w (cid:1) w = 0 , (3.17)which is the very BPS equation of the standard CP NL σ -model. Thus, a general solution of Eq. (3.15), up tothe gauge degrees of freedom, is given by w = P | P | , P = ( P ( z − ) , P ( z − ) , P ( z − )) T , (3.18)where P has no overall factor, and P a is a polynomial in z − . Therefore, we finally obtain the solution for the Z field Z = g − W = χg − w = χg − P , (3.19)where χ is a normalization factor. As the BPS bound (3.7) indicates, the lowest energy solution among Eq. (3.19) with a given backgroundfunction g possesses the highest topological charge. In terms of the explicit calculation of the topologicalcharge, we discuss the conditions for the lowest energy solutions.The topological charge (2.13) can be written in terms of Z as Q = − i π Z d x ε ij ( D i Z ) † D j Z . (3.20)We employ the constant background gauge field A + for simplicity. Then, the matrix g in Eq. (3.9) becomes g = exp ( − iA + z + ) , (3.21)so that the components of g − are given by power series in z + . It allows us to write Eq. (3.20) as a line integralalong the circle at spatial infinity Q = 12 π Z S ∞ C, (3.22)with C = − i Z † d Z [27, 38], since the one-form C becomes globally well-defined. To evaluate the integral inEq. (3.22), we write explicitly Z = χ p | P | + | P | + | P | X a g − a ( z + ) P a ( z − ) g − a ( z + ) P a ( z − ) g − a ( z + ) P a ( z − ) , (3.23)where g − ab is the ( a, b ) component of the inverse matrix g − .Let N a ( K ab ) be the highest power in P a ( g − ab ). Note that though g − ab are formally represented as powerseries in z + , the integers K ba are not always infinite; especially, if a positive integer power of A + is zero,all of K ba become finite because g − reduces to a polynomial of finite degree in z + . Using the plane polarcoordinates { r, θ } , one can write g − ba ( z + ) P a ( z − ) ∼ r N a + K ba exp[ − i ( N a − K ba ) θ ] at the spatial boundary andfind that only the components of the highest power in r contribute to the integral (3.22). Since we are interestedin constructing topological solitons, we consider the case when the physical space R can be compactified to thesphere S , i.e., the field Z takes some fixed value on the spatial boundary. Such a compactification is possible8igure 1: Topological charge density of the axial symmetric solution (3.28) with κ = 1 .if there is only one pair { N a , K ba } giving the largest sum N a + K ba or any pairs { N a , K ba } , sharing the largestsum, have the same value of the difference. For such configurations, the topological charge is given by Q = − N a + K ba , (3.24)where the combination { N a , K ba } yields the largest sum among any pairs { N c , K dc } . This equation (3.24)indicates that the highest topological charge configuration is given by the choice N a = 0 for a particular valueof a which gives the biggest K ba .We are looking for the lowest energy solutions with an explicit background field. As a particular example,let us consider A = κ ( λ + λ + λ ) , A = κ ( λ + λ − λ ) , (3.25)where κ is a constant. Clearly, this choice yields the potential term V = 4Tr ( n F ) = − √ κ n = 16 κ (cid:0) − h ( S ) i (cid:1) , (3.26)which can be interpreted as an easy-axis anisotropy, or quadratic Zeeman term, which naturally appears incondensed matter physics. In this case, the solution (3.19) can be written as Z = χ √ ∆ P ( z − ) + √ κz + e πi P ( z − ) P ( z − ) + iκz + P ( z − ) + κ z √ e πi P ( z − ) P ( z − ) . (3.27)Therefore, the solution with the highest topological charge is given by P = α , P = α z − + α with α i ∈ C ,and P being a nonzero constant. Choosing P = P = 0 , one can obtain the axially-symmetric solution Z = 1 √ ∆ √ κz + e πi κ z √ e πi , ∆ = 1 + 2 κ z + z − + κ z z − , (3.28)which possesses the topological charge Q = 2 . Note that this configuration also satisfies the BPS equationof the pure CP NL σ -model [26, 27, 31]. Figure 1 shows the distribution of the topological charge (3.20) ofthis solution (3.28) with κ = 1 . We find that the topological charge density has a single peak, although highercharge topological solitons with axial symmetry are likely to possess a volcano structure, see e.g., Ref. [39].These highest charge solutions give the asymptotic values at spatial infinity of the color field (cid:0) n ∞ , n ∞ , n ∞ , n ∞ , n ∞ , n ∞ , n ∞ , n ∞ (cid:1) = (0 , , − , , , , , / √ . (3.29)9igure 2: The expectation values h S a i for the solution (3.28) with κ = 1 .Figure 3: The expectation values h ( S a ) i for the solution (3.28) with κ = 1 .It indicates that n takes the vacuum value in the Cartan subalgebra of SU (3) . Hence, the vacuum of the modelcorresponds to a spin nematic, i.e., h S i = h S i = h S i = 0 and h ( S ) i = 0 , h ( S ) i = h ( S ) i = 1 . Unlikethe pure CP model, there is no degeneracy between the spin nematic state and ferromagnetic state in ourmodel because the SU (3) global symmetry is broken. As shown in Fig. 2, the spin nematic state is partiallybroken around the soliton because the expectation values h S a i become finite. Fig. 3 shows that h ( S a ) i of thesolution (3.28) are axially symmetric, although the expectation values h S a i have angular dependence. Note that the Hamiltonian (1.1) with B = 2 A admits closed-form analytical solutions [40]. Further, the CP BPS truncation corresponds to the restricted choice of the parameters, B = 2 A = κ . The relation B = 2 A is referred to as the solvable line, whereas the restriction B = 2 A = κ is called the BPS point [25]. Here weshow that similar restrictions occur in our model. For this purpose, we consider the generalized Hamiltonian H = H D + H L + H Boundary + ν H ani + µ H pot , (3.30)where ν and µ are real coupling constants. Here, H D indicates the CP Dirichlet term, i.e., the first term in ther.h.s of Eq. (2.10), and H L does the Lifshitz invariant term which is the second term of that. Explicitly, these10nd other terms read H D = ρ Z d x Tr ( ∂ k n ∂ k n ) , (3.31) H L = − iρ Z d x Tr ( A k [ n , ∂ k n ]) , (3.32) H ani = − ρ Z d x (cid:2) Tr (cid:0) [ A k , n ] (cid:1) − Tr (cid:0) [ A k , n ∞ ] (cid:1)(cid:3) , (3.33) H pot = 4 ρ Z d x [Tr ( n F ) − Tr ( n ∞ F )] , (3.34)where A k is a constant background field, as before. Finally, the boundary term H Boundary is defined by Eq. (3.1)with the 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 order to guarantee the finiteness of the total energy. Clearly, the Hamiltonian (3.30) is reduced toEq. (3.4) as we set ν = µ = 1 .The existence of exact solutions of the Hamiltonian (3.30) with ν = µ can be easily shown if we rescalethe space coordinates as ~x → r ~x , where r is a positive constant, while the background gauge field A k remainsintact. By rescaling, the Hamiltonian (3.30) becomes H = H D + r ( H L + H Boundary ) + r (cid:0) ν H ani + µ H pot (cid:1) . (3.35)Setting ν = µ and choosing the scale parameter r = ν − , one gets H r = ν − ν = µ = H D + ν − ( H L + H Boundary + H ani + H pot ) . (3.36)Notice that since the solutions (3.19) with P i being arbitrary constants are holomorphic maps from S to CP , they satisfy not only the variational equations δH ν = µ =1 = 0 but also the equations δH D = 0 , where δ denotes the variation with respect to n with preserving the constraint (2.6). Therefore, the solutions also satisfythe equations δH r = ν − ν = µ = 0 . This implies that, in the limit µ = ν , the Hamiltonian (3.30) supports a familyof exact solutions of the form Z ( ν ) = exp (cid:2) iν A + z + (cid:3) c , (3.37)where c is a three-component complex unit vector.Since the solution (3.37) is a BPS solution of the pure CP model with the positive topological charge Q ,one gets H D [ Z ( ν )] = 16 πρQ . In addition, the lower bound at the BPS point (3.7) indicates that H ν = µ =1 [ Z ( ν =1)] = − πρQ . Combining these bounds, we find that the total energy of the solution (3.37) is given by H ν = µ [ Z ( ν )] = 16 πρ (cid:18) − ν (cid:19) Q. (3.38)Since the energy becomes negative if ν < , we can expect that for small values of the coupling ν , the ho-mogeneous vacuum state becomes unstable, and then separated 2D Skyrmions (or a Skyrmion lattice) emergesas a ground state. In this section, we study baby Skyrmion solutions of the Hamiltonian (3.30) with various combinations of thecoupling constants. Apart from the solvable line, no exact solutions could find analytically, and then we have11o solve the equations numerically. Here, we restrict ourselves to the case of the background field given byEq. (3.25).For the background field (3.25), by analogy with the case of the single CP magnetic Skyrmion solution,we can look for a configuration described by the axially symmetric ansatz Z = (cid:0) sin F ( r ) cos G ( r ) e i Φ ( θ ) , sin F ( r ) sin G ( r ) e i Φ ( θ ) , cos F ( r ) (cid:1) , (4.1)where F and G ( Φ and Φ ) are real functions of the plane polar coordinates r ( θ ).The exact solution on the solvable line ν = µ with axial symmetry can be written in terms of the ansatzwith the functions F = tan − r ν κ r + ν κ r , G = tan − (cid:18) ν κr (cid:19) , Φ = θ + π , Φ = 2 θ + 3 π . (4.2)Further, the solution (3.28) is given by Eq. (4.2) with ν = 1 . This configuration is a useful referencepoint in the configuration space as we discuss below some properties of numerical solutions in the extendedmodel (3.30).For our numerical study, it is convenient to introduce the energy unit ρ and the length unit κ − , in order toscale the coupling constants. Then, the rescaled components of the Hamiltonian with the ansatz (4.1) become H D = Z d x (cid:2) F ′ + sin F G ′ ++ sin Fr n ˙Φ cos G + ˙Φ sin G o − sin Fr (cid:16) ˙Φ cos G + ˙Φ sin G (cid:17) (cid:21) , (4.3) H L = − Z d xr " √ (cid:16) θ + π − Φ (cid:17) ( r (cid:18) cos GF ′ − sin 2 F sin G G ′ (cid:19) + sin 2 F cos G ˙Φ − sin 2 F sin F cos G (cid:16) cos G ˙Φ + sin G ˙Φ (cid:17) o − sin ( θ + Φ − Φ ) (cid:26) r sin F G ′ + 12 sin F sin 2 G (cid:16) ˙Φ + ˙Φ (cid:17) − sin F sin 2 G (cid:16) cos G ˙Φ + sin G ˙Φ (cid:17) o , (4.4) H ani = 12 Z d x "
16 sin F cos G (cid:26) cos F − √ (cid:16) − Φ + π (cid:17) sin 2 F sin G + sin F sin G (cid:27) + sin F (1 + 2 sin G ) + 8(cos F − cos G sin F ) + 4 cos G sin F − , (4.5) H pot = 2 Z d x (cid:16) − √ n (cid:17) = 6 Z d x cos F, (4.6)where the prime ′ and the dot ˙ stands for the derivatives with respect to the radial coordinate r and angu-lar coordinate θ , respectively. The system of corresponding Euler-Lagrange equations for Φ i can be solvedalgebraically for an arbitrary set of the coupling constants, and the solutions are Φ = θ + π , Φ = 2 θ + 3 π mπ , (4.7)12here m is an integer. Without loss of generality, we choose m = 0 by transferring the corresponding multiplewindings of the phase Φ to the sign of the profile function G . Then, the system of the Euler-Lagrange equationsfor the profile functions with the phase factor (4.7) reads δH D δF + δH L δF + ν δH ani δF + µ δH pot δF = 0 ,δH D δG + δH L δG + ν δH ani δG + µ δH pot δG = 0 , (4.8)with δH D δF = (cid:20) rF ′′ + 2 F ′ − sin 2 F (cid:26) rG ′ + 1 + 3 sin Gr − Fr (cid:0) G (cid:1) (cid:27)(cid:21) , (4.9) δH L δF = − h √ F {− r sin GG ′ + cos G + cos G (cid:0) G (cid:1) (cid:0) F − (cid:1) }− r sin 2 F G ′ −
32 sin 2 F sin 2 G + 4 cos F sin F sin 2 G (cid:0) G (cid:1)(cid:21) , (4.10) δH ani δF = 2 r h √ G cos G sin F (cid:0) − F (cid:1) − F sin F cos G +4 sin 2 F (cid:8) cos F − sin F cos G (cid:0) G (cid:1)(cid:9) − sin 2 F cos 2 F (1 + 2 sin G ) (cid:3) , (4.11) δH pot δF = 6 r sin 2 F, (4.12) δH D δG = (cid:20) r sin F G ′′ + 2 r sin 2 F F ′ G ′ + 2 sin F G ′ − sin F sin 2 Gr (cid:8) − F (cid:0) G (cid:1)(cid:9)(cid:21) , (4.13) δH L δG = − h √ F sin G (cid:8) rF ′ + sin 2 F (cid:0) − G (cid:1)(cid:9) + r sin 2 F F ′ + sin F (1 − G ) + sin F (cid:0) G − G (cid:1)(cid:3) , (4.14) δH ani δG = r h √ F sin F cos G (cid:0) − G (cid:1) + 16 sin F cos G sin G − sin F sin 2 G i , (4.15) δH pot δG = 0 . (4.16)We solve the equations for ν = µ numerically with the boundary condition F (0) = G (0) = 0 , lim r →∞ F ( r ) = lim r →∞ G ( r ) = π/ , (4.17)which the exact solution (4.2) satisfies. This vacuum corresponds to the spin nematic state (3.29).Let us consider the asymptotic behavior of the solutions of the equations (4.8). Near the origin, the leadingterms in the power series expansion are F ≈ c F r, G ≈ c G r, (4.18)where c F and c G are some constants implicitly depending on the coupling constants of the model. To see thebehavior of solutions at large r , we shift the profile functions as F = π − F , G = π − G . (4.19)13igure 4: Plot of the profile functions { F, G } (left) and the topological charge density (right) of numericalsolutions for changing the coupling constant ν at µ = 1 . . The gray line indicates the quantities of the exactsolution (4.2) on the solvable line. ν H H D H L ν H ani µ H pot H Boundary
Derrick Q µ = 1 . where ”Derrick”denotes the value ( H L + H Boundary ) / ( ν H ani + µ H pot ) , which is expected to be − by the scaling argument.For ν = 1 . , we used the exact solution (4.2) so that the ”Derrick” and topological charge for ν = 1 . areexact values.Then, one obtains linearized asymptotic equations on the functions F and G of the forms (cid:18) F ′′ + F ′ r − F r (cid:19) + 2 √ (cid:18) G ′ − G r (cid:19) − (cid:0) ν + 3 µ (cid:1) F = 0 , (cid:18) G ′′ + G ′ r − G r (cid:19) − √ (cid:18) F ′ + 2 F r (cid:19) = 0 . (4.20)Unfortunately, the equations (4.20) may not support an analytical solution. However, these equations implythat the asymptotic behavior of the profile functions is similar to that of the functions (4.2), by a replacement ν κ with ( ν + 3 µ ) / . Indeed, the asymptotic equations (4.20) depend on such a combination of the couplingconstants, and there may exist an exact solution on the solvable line with the same character of asymptoticdecay as the localized soliton solution of the equation (4.8).To implement a numerical integration of the coupled system of ordinary differential equations (4.8), weintroduce the normalized compact coordinate X ∈ (0 , via r = 1 − XX . (4.21)The integration was performed by the Newton-Raphson method with the mesh point N MESH = 2000 .In Fig. 4, we display some set of numerical solutions for different values of the coupling ν at µ = 1 . and their topological charge density Q defined through Q = 2 π R r Q d r . The solutions enjoy Derrick’s scaling14elation and possess a good approximated value of the topological charge, as shown in Table 1. One observesthat as the value of the coupling ν becomes relatively small, the function G is delocalizing while the profilefunction F is approaching its vacuum value everywhere in space except for the origin. This is an indicationthat any regular non-trivial solution does not exist ν = 0 . Asymptotic interaction of solitons is related to the overlapping of the tails of the profile functions of well-separated single solitons [3]. Bounded multi-soliton configurations may exist if there is an attractive forcebetween two isolated solitons.Considering the above-mentioned soliton solutions of the gauged CP NL σ -model, we have seen thatthe exact solution (4.2) has the same type of asymptotic decay as any solution of the general system (4.8).Therefore, it is enough to examine the asymptotic force between the solutions on the solvable line (4.2) to un-derstand whether or not the Hamiltonian (3.30) supports multi-soliton solutions of higher topological degrees.Thus, without loss of generality, we can set µ = ν .Following the approach discussed in Ref. [3], let us consider a superposition of two exact solutions above.This superposition is no longer a solution of the Euler-Lagrange equation, except for in the limit of infiniteseparation, because there is a force acting on the solitons. The interaction energy of two solitons can be writtenas E int ( R ) = H sp ( R ) − H exact , (4.22)where H sp ( R ) is the energy of two BPS solitons separated by some large but finite distance R from each other,and H exact stands for the static energy of a single exact solution. Notice that the lower bound of the Hamiltonian(3.30) with µ = ν is given H = ν − H ν = µ =1 + (1 − ν ) H D ≥ π (1 − ν − ) Q, (4.23)where the equality is enjoyed only by holomorphic solutions. Therefore, we immediately conclude H sp ( R ) ≥ H exact , (4.24)where the equality is satisfied only at the limit R → ∞ . It follows that the interaction energy is always positivefor finite separation, and the interaction is repulsive. Since the exact solution has the topological charge Q = 2 ,it implies that there are no isolated soliton solutions with the topological charge Q ≥ in this model. Notethat, however, as the BPS solution (3.19) suggests, there can exist soliton solutions with an arbitrary negativecharge, which are topological excited states on top of the homogeneous vacuum state. In this paper, we have studied two-dimensional Skyrmions in the CP NL σ -model with a Lifshitz invariantterm which is an SU (3) generalization of the DM term. We have shown that the SU (3) tilted FM Heisenbergmodel turns out to be an SU (3) gauged CP NL σ -model in which the term linear in a background gauge fieldcan be viewed as a Lifshitz invariant. We have found exact BPS-type solutions of the gauged CP model inthe presence of a potential term with a specific value of the coupling constant. The least energy configurationamong the BPS solutions has been discussed. We have reduced the gauged CP model to the (ungauged) CP model with a Lifshitz invariant by choosing a background gauge field. In the reduced model, we haveconstructed an exact solution for a special combination of coupling constants called the solvable line andnumerical solutions for a wider range of them. 15or numerical study, we chose the background field, generating a potential term that can be interpreted asthe quadratic Zeeman term or uniaxial anisotropic term. One can also choose a background field generating theZeeman term; if the background field is chosen as A = − κλ and A = κλ , the associated potential term isproportional to h S i . The Euler-Lagrange equation for the extended CP model with this background field isnot compatible with the axial symmetric ansatz (4.1). Therefore, a two-dimensional full simulation is requiredto obtain a solution with this background field. This problem, numerical simulation for non-axial symmetricsolutions in the CP model with a Lifshitz invariant, is left to future study. In addition, the construction of a CP Skyrmion lattice is a challenging problem. The physical interpretation of the Lifshitz invariants is also animportant future task. The microscopic derivation of the SU (3) tilted Heisenberg model [21] may enable us tounderstand the physical interpretation and physical situation where the Lifshitz invariant appears. Other futurework would be the extension of the present study to the SU (3) antiferromagnetic Heisenberg model wheresoliton/sphaleron solutions can be constructed [41–43].We restricted our analysis on the case that the additional potential term µ H pot is balanced or dominantagainst the anisotropic potential term ν H ani , i.e., ν ≤ µ . We expect that a classical phase transition occursoutside of the condition, and it causes instability of the solution. At the moment, the phase structure of themodel (3.30) is not clear, and we will discuss it in our subsequent work.Moreover, it has been reported that in some limit of a three-component Ginzburg-Landau model [44, 45],and of a three-component Gross-Pitaevskii model [46, 47], their vortex solutions can be well-described byplanar CP Skyrmions. We believe that our result provides a hint to introduce a Lifshitz invariant to themodels, and that our solutions find applications not only in SU (3) spin systems but also in superconductorsand Bose-Einstein condensates described by the extended models, including the Lifshitz invariant. Acknowledgments
This work was supported by JSPS KAKENHI Grant Nos. JP17K14352, JP20K14411, and JSPS Grant-in-Aidfor Scientific Research on Innovative Areas “Quantum Liquid Crystals” (KAKENHI Grant No. JP20H05154).Ya.S. gratefully acknowledges support by the Ministry of Education of Russian Federation, project FEWF-2020-0003. Y. Amari would like to thank Tokyo University of Science for its kind hospitality.
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