Analytical Skyrmion Solutions of the Non-Linear Sigma Model S

Vol. 135 (2019) ACTA PHYSICA POLONICA A No. 6

Proc. 13th International School on Theoretical Physics: Symmetry and Structural Properties of Condensed Matter Analytical Skyrmion Solutions of the Non-Linear Sigma Model S. Trimper∗ Institute of Physics, Martin-Luther-University, Halle, Germany Motivated by studying quantum topological states we analyze the mesoscopic non-linear sigma model in two dimensions. We find systematically analytical solutions for the spatially depending magnetization density for arbitrary winding number under the constraint of topological protection. The methods allows also to get analytical expressions for other alignments such as a quadrupolar one. The inhomogeneous energy and skyrmion densities are calculated. Using a formulation of the local magnetization density in terms of a two component complex spinor field we get the related vector potential playing the role of the Berry curvature. The related magnetic field offers a spatial dependence which is completely different from electrodynamics. The dynamical stability of the solutions are discussed. DOI: 10.12693/APhysPolA.135.1275 PACS/topics: magnetism, non-linear sigma model, topological states, skyrmions

1. Introduction Notice that chirality in form of a term n · ∇ × n [10–12] is not included in our model. Because of Skyrmions are stable field configurations which are n(x, −t) = −n(x, t) the energy density h(x, t) is protected by a topological constraint. Originally the con- invariant under time reflection. The constraint n2 = 1 cept had been introduced by Skyrme [1] in the context implies a local sphere in spin space, the total surface of of particle physics following Heisenberg’s idea to identify which is a topologically protected invariant particles as inhomogeneous field configurations with fi- 1 Z 1 Z Q = d2xn · (∂ n × ∂ n) = d2xq(x). (2) nite and localized energy density. In a previous paper we 4π x y 4π have considered such a situation for the non-linear sigma Here we have introduced the density of the topological model [2] based on [3]. A similar procedure has been invariant (skyrmion density) q(x) = q(x, y) on the x–y applied by [4] and in more detail by [5, 6] for a review plane. The topological invariant Q is changed under comparison [7]. The motivation of the present note is time reflection Q(−t) = −Q(t) whereas the energy is to offer a systematic way in getting analytical topologi- invariant under time reflection. Moreover, Q changes its cally protected skyrmion configurations for the isotropic sign in case the coordinates x and y are interchanged sigma model without any additional chiral terms. Using a whereas the energy density remains invariant. This fact representation of the magnetization density in terms of a leads to a degeneration of skyrmion solutions discussed complex spinor field [8] the system reveals a gauge poten- later. The Hamiltonian is altered under local rotation tial which adopts the Berry connection well established n(x) → n(x) + Θ(x) × n(x) to in quantum systems. The related magnetic field playing Z the role of the Berry curvature can be calculated for cer- d α α δH=J d xΠµ ∂µΘα, with Πν =εαβγ nβ∂ν nγ . (3) tain skyrmion alignments such as a configuration with α arbitrary winding number and for instance a quadrupo- The tensor Πµ , α = 1, . . . , n = 3, µ = 1, . . . , d = 2 is lar arrangement. Moreover, we discuss the dynamical the spin-current density which appears due to the local stability of the analytical solutions. distortion. In terms of Π µ = J(n × ∂µn) the topological density in Eq. (2) can be expressed by 1 1 2. Sigma model and gauge potential q(x) = (∂ n )(ε Π α) ≡ (∂ n )Γ α. (4) 2 µ α νµ ν 2 µ α µ α The nonlinear-sigma model is a classical continu- Here the quantity Γµ is related to the spin current and ous version of the quantum Heisenberg model. It is fulfills 2 α 2 formulated in terms of a local real three component (∂µnα) = (Γµ ) . (5) magnetization field n(x, t) with a fixed length which is To reflect the properties of the underlying spins as related to the conservation of the total spin [9]. The complex quantities and to separate the fields and the Hamiltonian reads in two-dimensional case local gauge potential let us introduce [8, 10]: Z J H = d2xh(x), h(x) = (∇n)2. (1) n = ψ†σ ψ, ψ† = (ψ∗, ψ∗). (6) 2 α α 1 2 Here σα are the (complex) Pauli-matrices whereas ψ(x) is a two-component complex spinor field with the properties |ψ|2 = 1, → ψ∂ ψ† = −ψ†∂ ψ. (7) ∗e-mail: [email protected] µ µ

(1275) 1276 S. Trimper

X y − yq After a straightforward calculation the Hamiltonian θ(x, y) = q tan−1 , q = ±1, ±2 ... (20) x − x in Eq. (1) reads q q Z The integer factor q, the winding number, appears be- H = 2J (|∂ ψ|2 − A2 )d2x, (8) µ µ cause θ is an angle. Inserting this solution in Eq. (19) where the vector gauge potential A is given by one finds the solution for the azimuthal angle ϕ as ∓q i Y rq A = − [ψ∗∂ ψ − ψ ∂ ψ∗] ≡ −iψ†∂ ψ. (9) ϕ(x, y) = tan−1 . (21) µ 2 i µ i i µ i µ a In the quantum context the potential A is called the q The solution given in Eqs. (20), (21) had been already ob- Berry connection, for a recent review see [13]. The re- 2 2 lated magnetic field for d = 2 is given by tained in [2]. Here a is a lower cut-off, rq = (x − xq) + (y − y )2 with the constants x , y which will be identi- B = ∂ A = ∂ A − ∂ A . (10) q q q z µν µ ν x y y x fied with position of the skyrmions. The solution reflects This field Bz corresponds to the Berry curvature. Un- the already mentioned topological degeneracy, that the der the local gauge transformation of the complex field interchange y ↔ x is related to altering q ↔ −q After ψ with a real function Λ(x) according to gauge transformation Eq. (12) the Berry connection is 0 i Λ(x) 2 ψi → ψi = e ψi, (11) Aµ = (∂µθ) cos ϕ. (22) the vector potential Aµ is transformed to 0 4. Solution with arbitrary winding number q Aµ = Aµ(ψ) + ∂µΛ. (12) Introducing the covariant derivative Dµ = ∂µ − iAµ one can express H and the topological invariant as Let us illustrate the approach for an arbitrary q which Z leads due to Eqs. (20) and (21) to H = 2J ddx(D ψ )∗(D ψ ), y r ∓q µ i µ i θ(x, y) = q tan−1 , ϕ(x, y) = tan−1 . (23) i Z x a Q = d2x D A (13) Here r2 = x2 + y2 is the distance from the position 2π µν µ ν of the skyrmion, the core of which is situated at the Due to the gauge transformation the number of degrees origin x = y = 0. In terms of polar coordinates of freedom remains unchanged. q q x = r cos α, y = r sin α in the coordinate space d = 2 the order parameter field is 3. Skyrmion solutions in d = 2 2(ra)q n (r, α) = cos(qα), x r2q + a2q A local distortion of the magnetization field n(x) is re- 2(ra)q lated to both, a change of the energy density h(x) and si- ny(r, α) = − sin(qα), r2q + a2q multaneously to a change of the topological density q(x). To reveal a relation between h(x) and q(x) let us con- r2q − a2q n (r, α) = ± → ±1 for r → ∞. (24) sider the inequality z r2q + a2q α 2 R(x) = (∂µnα ∓ Γµ ) ≥ 0. (14) As an example the magnetization field is depicted in Using Eqs. (4), (5) the local condition follows Fig. 1 for q = 1. At the position of the origin of the skyrmion at x = y = h(x) ∓ Jq(x) ≥ 0. (15) 0 the magnetization field is n(0, 0) = (0, 0, −1) for the This relation is a local one indicating the above men- upper sign in Eq. (17) and n(0, 0) = (0, 0, +1). As indi- tioned strong correlation between the local energy den- cated in Eq. (24) the spin flips to n(r → ∞) = (0, 0, ±1). sity and the local skyrmion density. In case of d = 2, The topological density according to Eq. (2) reads i.e. the magnetization field is in a plane, such as x–y q2a2qr2q−2 plane, we integrate the last relation finding the global q(r) = ± . (25) relationship H ≥ ±4πJQ. From here we conclude the π(r2q + a2q)2 global condition for the minimal energy of the topologi- The integration leads to Q = R q(r)drrdα = ±q. The cally protected states skyrmion density q(x) and simultaneously the energy density offers a pronounced peak around the origin which Hmin = 4πJ|Q|. (16) is shown in Fig. 2. The last relation is realized for R(x) = 0 in Eq. (14): α In the same manner we find the gauge potential A ∂µnα = ±Γµ . (17) which gives rise to the Berry curvature. Using the representation 2q2(ar)2q i θ1 i θ2 B = ∂ A − ∂ A = ± . (26) ψ1 = e cos ϕ, ψ2 = e sin ϕ. (18) z x y y x r2(r2q + a2q)2 Equation (17) is consistent with 2 −2(q+1) The field Bz ' q r is totally different from con- 2∂xϕ = ∓ sin 2ϕ∂yθ, 2∂yϕ = ± sin 2ϕ∂xθ. (19) ventional electrodynamics. The related current rotB = j˜ From here we conclude that the phase θ = θ1 − θ2 in the offers in polar coordinates a nonzero component in angle realization in Eq. (18) fulfills ∇2θ = 0 with the solution direction Analytical Skyrmion Solutions of the Non-Linear Sigma Model 1277

Fig. 3. Quadrupolar conﬁguration with alternating Fig. 1. A single skyrmion with q = 1. winding numbers q = ±1.

Fig. 2. Skyrmion density q(x, y) on the x–y plane with Fig. 4. Skyrmion density q(x, y) on the x–y plane for winding number q = 1. a quadrupolar conﬁguration.

4q2(ar)2q 5. Dynamics ˜j = ± (1 + q)r2q + (1 − q)a2q . (27) α r3[r2q + a2q]3 This current is directly related to the topological current In this section we discuss the dynamical stability of the introduced in Eq. (32). We have discussed a lot of dif- skyrmion solution. In case the magnetization field fulfills ferent analytical solutions such as dipolar configuration the Landau–Lifshitz–Gilbert equation (n) (n) with q = ±1 or with both q = 1, different linear align- ∂tn + J∂ν Π ν = −Ω , withΩ = +γ, ˙n × n. (30) ments, hexagonal configurations and other ones. Here we The continuity equation for the density u(x, t) = h(x, t) present the quadrupolar order with 4 alternating wind- or q(x, t) reads ing numbers. Two skyrmions with q = 1 and distance 2a ∂u(x, t) are along the x-axis whereas two skyrmions with q = −1 + divj(u) = −Ω (u) (31) along the y-axis and likewise distance 2a. The analytical ∂t (e) (q) solution is with Ω = Jαn · (∇n × ∇n˙ ), Ω = Jα(∇n˙ × ∇n)z. r4 − a4 4a2xy The related currents are nx = , ny = − , (q) (h) α r4 + a4 r4 + a4 jµ = Jαβ∂αnγ ∂µβnγ .jµ = Jαβγ ∂µn nβ∂νν nγ . (32) 2a2(y2 − x2) n = ± . (28) For arbitrary q the topological current is in polar coordi- z r4 + a4 nates The solution is shown in Fig. 3. 4(ra)2qq2 (q + 1)r2q + (1 − q)a2q The B field is isotropic and simply given by B = (q) z j (x) = 2J 3 2q 2q 3 eα, ±8a4r2(r4 + a4)−2. The density q(x) is given by r (r + a ) 16a4r2 (33) q(x) = ,Q = 2. (29) (r4 + a4)2 which is up to a factor the above introduced fictitious current j˜ (see Eq. (27)). Insofar the topological current and is shown in Fig. 4. is directly related to the Berry field according to Eq. (27). 1278 S. Trimper

The topological current is free of sources, i.e. ∇·j(q) = 0. References Due to the continuity equation Eq. (31) for zero damping this fact reflects the dynamical stability of the skyrmion [1] T.H.R. Skyrme, Nucl. Phys. 31, 556 (1962). solutions. In the same manner we get for the solution [2] S. Trimper, Phys. Lett. A 70, 114 (1979). α [ ] A.A. Belavin, A.M. Polyakov, JETP Lett. 22, 245 with arbitrary q the relation ∂µΠµ = 0 which is accor- 3 dance with Eq. (30) without zero damping. Again the (1975). dynamical stability of skyrmion configurations is guaran- [4] A.N. Bogdanov, D.A. Yablonskii, JETP 68, 101 teed. The inclusion of damping effects will be discussed (1989). elsewhere. [5] S. Mülbauer, B.B.F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, P. Bni, Science 323, 915 (2009). 4. Conclusion [6] N. Nagaosa, Y. Tokura, Nat. Nanotechnol. 8, 899 (2013). We have analyzed the mesoscopic non-linear sigma model in order to find topologically protected inhomo- [7] G. Tatara, Physica E 106, 208 (2019). geneous solutions for the magnetization field, the energy, [8] N. Nagaosa, Quantum Field Theory in Strongly Cor- and the skyrmion density. The method can be extended related Electronic Systems, Springer, New York 1999. to get more refined skyrmion configurations. Moreover, [9] A.M. Polyakov, Phys. Lett. B 59, 79 (1975). we have discussed the dynamical behavior of the solution. [10] J.H. Han, J. Zang, Z. Yang, J.H. Park, N. Nagaosa, The solutions presented are dynamically stable. Phys. Rev. B 82, 094429 (2010). [11] W.T. Hou, J.X. Yu, M. Daly, J. Zang, Phys. Rev. B 96, 140403 (2017). Acknowledgments [12] H. Shimada, K. Takahashi, H.T. Ueda, Phys. Rev. B 97, 224424 (2018). I gratefully acknowledge discussions with Ingrid Mer- tig, Alexander Mock, and Börge Göbel. [13] S. Rachel, Rep. Prog. Phys. 81, 116501 (2018).