JHEP02(2021)095 Springer November 5, 2020 February 11, 2021 December 24, 2020 : : : Received Published Accepted , a Published for SISSA by https://doi.org/10.1007/JHEP02(2021)095 and Muneto Nitta a [email protected] , . 3 2003.07147 Norisuke Sakai The Authors. 1 c Integrable Field Theories, Solitons Monopoles and Instantons

a,b, We study two-body interactions of magnetic skyrmions on the plane and apply , [email protected] Corresponding author. 1 Department of Physics, UniversityCork, College Ireland Cork, E-mail: [email protected] Department of Physics and Research andHiyoshi Education 4-1-1, Center Yokohama, for Kanagawa Natural Sciences, 223-8521, Keio Japan University, b a Open Access Article funded by SCOAP transition between the skyrmion lattice and an inhomogeneous spiralKeywords: state. ArXiv ePrint: ferromagnetic state is noexactly longer the without lowest the energyinteraction state. ambiguities energy This of for critical numerical a valueto pair is simulations. the determined of exponential skyrmions decay Along is ofexpressions the a repulsive we purely solvable with construct Zeeman power line an potentialground law the inhomogeneous term. fall states skyrmion Using off for the lattice in interaction state, the contrast energy which model is in a particular candidate parameter regions. Finally we estimate the Abstract: them to a (mostly)of analytic the description solvable of line, a aterms, skyrmion particular where lattice. choice analytic of This expressions a for issingle skyrmions potential done are for skyrmion in available. magnetic the solutions The anisotropy context energy is and of Zeeman found these analytic to become negative below a critical point, where the Calum Ross, Skyrmion interactions and latticesanalytical in results chiral magnets: JHEP02(2021)095 ]. 9 , ]. In 8 5 – )[ 3 κ ]. The skyrmion lattice phase 14 14 – ] magnetic skyrmions are stabilised 11 7 , 6 ]. A systematic mathematical study of 7 4 12 19 10 12 ] as minimum energy configurations of the mag- – 1 – 9 22 2 6 ]. A model of a skyrmion lattice made up of non 16 10 16 , 15 14 4 1 23 18 ]. The various phases of a chiral magnet have been explored with easy plane ]. They were first predicted in [ 10 1 4.2 Transition to the spiral state 3.1 Superpositions3.2 of skyrmions Qualitative analysis3.3 of interactions Computation of the interaction energy 4.1 Average energy density of skyrmion lattice state 2.1 The model 2.2 Exact single2.3 skyrmion solutions Instability of ferromagnetic phase diction with the usualmagnetic Derrick skyrmions scaling in argument the [ is presence given of in an [ externalanisotropy magnetic in field addition and to withouthas an anisotropy been external studied magnetic extensively field in [ [ materials [ netisation vector field and havecontrast been to the observed case in of a babystabilised variety skyrmions by of — higher field magnetic derivative theoretical materials terms analoguesby [ in of a the magnetic energy skyrmions first [ orderThe fact Dzyaloshinskii-Moriya that (DM) the interaction DM term term can (with give coefficient a negative contribution to the energy avoids a contra- 1 Introduction Magnetic skyrmions are topologically non-trivial solitons which occur in certain magnetic B Expression for the energy density C Spiral state 5 Conclusion A DM interaction and boundary terms 4 Skyrmion lattice states 3 Interactions of magnetic skyrmions Contents 1 Introduction 2 Solvable models for magnetic skyrmions JHEP02(2021)095 , κ A κ 2 has ]. , one A A ]. 29 B > – = 2 19 27 = 2 , B 18 B ] a specific potential, a 20 ]. Spiral or helical states , and the DM constant, A 17 , can be tuned. This suggests , numerical analysis is needed . The slower, power law, decay B ], almost all theoretical studies A A 5 2 2 – 3 B > B > ], in particular in supersymmetric field 7 ) and an easy plane term (with coefficient B . This is a so-called Bogomol’nyi-Prasad- 2 κ of parameters is called the solvable line. These – 2 – = A A ]. While the anisotropy, , there is a special point tuned to the coefficient = 2 22 A = 2 B B = 2 B ] point where the model has an infinite number of analytic ]). In the case of superconductors, it corresponds to the criti- 24 , 26 , ]. The holomorphic properties of the skyrmion solutions along the 23 25 21 ]. This boundary contribution is closely related to the slow power law . This relation 1 − 31 , . Outside the solvable line, such as for = A 2 30 , Q ) 20 2 B > S The asymptotic decay of the skyrmion fields is also a key difference with a power law de- The exact skyrmion solution along the solvable line reveals the subtlety that a boundary Along the solvable line Apart from ground breaking experimental studies [ → ), was considered which results in a solvable model for skyrmions in chiral magnetic thin 2 cay for the solvable line and anfor exponential skyrmions decay along for the solvableare line not as mean easily that applicable. theexponential This asymptotic decay, is play methods because an used boundary important role terms for the along in subtlety the the of solvable energy, boundary which line. terms, vanish Even for the though presence of exact solutions overrides this disadvantage. such as skyrmion lattice states, ifskyrmions a balance and is the met between interaction thesingle energy negative skyrmion energy solutions between of are individual them. only knownregion It for the is solvable line worth and emphasisingto not for construct that the skyrmion exact better solutions. studied defined [ asymptotic decay of thecontribution magnetisation can give vector a for negative thewhich contribution can to skyrmion result the solution. in energy the ofstate. skyrmion This the having exact This boundary lower fact skyrmion energy opens solution, than up the the homogeneous possibility ferromagnetic of understanding the inhomogeneous ground state, theories (see, e.g., [ cal coupling between type-I andmagnetic type-II skyrmions superconductivities. and instantons Some in further chiral studies magnets of have BPS beencontribution made to in the refs. energy [ functional is required for the DM interaction term to be well of the DM interactionSommerfield through (BPS) [ solutions constructed in termscussed of in the holomorphic context functions. of monopolesin Such in various quantum BPS theories field admitting limits theories, topological and were are solitons first now [ known dis- to appear solvable line were first discussedare in determined [ by the materialthat the external the magnetic solvable field, line shouldexternal be magnetic accessible field. in real materials through a careful tuning of the A films. More explicitly,obtains tuning exact hedgehog the skyrmion twoS solutions coefficients with of topological the chargeexact (degree skyrmion potential solutions as to are a a be mapping Heisenberg natural generalisation model of the [ ferromagnetic skyrmions in the have also been studied as effectively one-dimensional inhomogeneous statesof [ magnetic skyrmionsunderstanding are was based lacking. on Oncombination the numerical of other a simulations Zeeman hand, term thus recently (with far in coefficient ref. and [ more analytic interacting closely packed skyrmions has been considered in [ JHEP02(2021)095 ] 19 , . This 18 A 2 ] becomes ]. The case ] along the 17 36 (for magnetic 11 B > A ], only considers ], it is known that 35 , 12 34 ]. This approximation , to understand some of 6 11 A ], using the dipole approxi- 35 – becomes large. For small values 33 A , and the two parameters κ – 3 – ]. Most previous work, including refs. [ 6 correspond to homogeneous ferromagnetic phases. The solvable B ] and has recently been discussed [ (for a Zeeman field) in the potential. From the numerical study of the 32 and B A ]. The other possible inhomogeneous state is a spiral or helical state [ , lies along the phase boundary between the polarised ferromagnetic phase B 16 , = 15 ] where an attractive interaction was found between individual skyrmions. The A 2 37 , on the other hand, the skyrmions along the solvable line have negative energy and We apply the dipole approximation at smaller values of In this paper we conduct a semi-analytic study of interactions between magnetic The interaction energy between a pair of skyrmions was discussed for a specific easy- A becomes larger. Thus the dipole approximation, at a fixed separation between two iso- as a function ofbe their used separation. reliablyphase. to Our find However, results the the show transitiontransition dipole that between between approximation the the the lattice breaks dipole ferromagneticof phase down a approximation phase and before skyrmion the can and lattice we spiral as, the can phase. effectively, lattice non study We interacting find the closely evidence phase packed that skyrmions a [ model the ground state beingstudies an [ inhomogeneous skyrmionwhere lattice the in configuration agreement is with effectively one numerical dimensional. the qualitative behavioursolvable of line. the We take triangular two exact skyrmion single skyrmion lattice solutions and phase consider from their superposition [ ferromagnetic ground state. TheA skyrmions have smaller radiilated as skyrmions, the works potential best parameter asof the potential parameter the homogeneous ferromagnetic state becomes unstable. This leads to the possibility of phase diagram of alarge chiral values magnet of withline, easy plane anisotropy, [ and the spin cantedable ferromagnetic line, phase. magnetic At skyrmions large are values obtained of as the positive potential energy along excited the states solv- on top of the stituent skyrmions becomes lessbe pronounced. measured The in accuracy termsof of of a the the single approximation separation skyrmion.ful should between parameters, two The the skyrmions diameter DM dividedanisotropy) of interaction by and strength the the diameter skyrmion is determined by three dimension- skyrmions along the solvable line,To and achieve apply this, it we toinvolves make construct working a use model with of of the askyrmions a dipole skyrmion superposition are lattice. approximation well of from separated, two [ becoming skyrmions, less and reliable is as reliable the distinction as between long the as con- the ref. [ skyrmion interaction energy forbefore. the The boundary solvable contributions linethe to does the skyrmion energy not configurations caused complicate appear bynecessitates the the to a slow, asymptotic slightly have power approach different law, been used approach decayskyrmion studied of for which solutions. is possible due to the presence of analytic axis potential in [ mation introduced in ref.the [ case of an externalthe magnetic interaction energy field decays without exponentially anof at anisotropy easy large term. plane separations It in anisotropy has this being been case more found [ important that than the Zeeman term was considered in JHEP02(2021)095 . , ]. C 36 2.1 (2.1) – 34 gives the A ] the following , the interaction , we see that the 15 4 , we calculate the , x, A 2 3 d 11 ,  2 ) 3 n ( we express the interaction U B ] for the case of a specific easy- ) + 32 ~n × α − , we review solvable models for magnetic ∇ 2 ( · . In works such as [ κ ~n – 4 – = 1 + 2 2 ~n , we present exact skyrmion solutions. Finally in | ]. We show that the power law decay of the skyrmion ~n 38 |∇ 2.2 1 2  2 R Z = is devoted to a summary and discussion. Appendix , constrained as bulk 5 2 ] ~n S [ E → 2 , we discuss the instability of the ferromagnetic phase. R : 2.3 ~n This paper is organized as follows. In section The key feature of the present work is that by working on the solvable line there are in two spatial dimensions describedtor in field terms of aenergy real functional three is component considered magnetisation vec- subsection 2.1 The model This subsection gives a concisevant summary for of our previously study known and results clarifies that our are conventions directly at rele- the same time. We study chiral magnets 2 Solvable models forIn magnetic this skyrmions section,we we introduce review the model solvablebe and models classify neglected. for homogeneous magnetic ground Then states, skyrmions. in when subsection the In DM subsection term can energy is applied to describe ais skyrmion estimated. lattice, and the Section phasedetails transition to of the the spiral phase boundarybehaviour term of in the theenergy magnetisation energy as functional vector a and field. function its of relation dimensionless In variables. to appendix Spiral the states asymptotic are described in appendix have a well defined variationalsolvable problem line, when we deriving find the a equations repulsive of interaction. motion. Alongskyrmions the and clarify theinteraction energy necessity and of force a between two boundary isolated term. skyrmions. In In section section axis potential. Comparinginteraction these energy previous will studies exponentiallyTwo-body decay with and when our the three-body appendix skyrmion skyrmionthe field interactions potential decays in have exponentially. been the studied absencefields in of [ necessitates an the anisotropy inclusion term of in a boundary term in the energy functional in order to we have a state lowerskyrmion in configurations energy with than the dipole thefor approximation ferromagnetic the we state. interaction arrive energy at Combining density an theically. of analytic exact the After expression superposition numerical single integration without we thepower find need law that to decay the in work interaction asymptot- contrast energy toAn is the exponentially exponential positive decaying decay and interaction of has was the a also pure found Zeeman in potential [ [ field where the transition to the spiral phase occurs. exact expressions for theabout skyrmion the configurations energy of enablingvalue a us of single to the skyrmion. anisotropy make This below exact which makes statements the it single possible skyrmion for solution us has to negative identify energy a and specific a better approximation. Using this we estimate the critical value of the external magnetic JHEP02(2021)095 (2.3) (2.2) . : 2 3 An A, B as coefficient, + 3 B Bn − ) = 3 2 n A ( 2 U , for which the DM term can , κ .    1 2 2 . |  3 2 ). α∂ depending on values of ), B 1 1 α∂ | ), A ) , An B − | 2 3 corresponds to the Bloch type DM and n A + = | ( = + cos 3 0 + sin 3 = +1 U 1 3 = 0 n 1 3 n Bn n A 2 α∂ is the Dzyaloshinskii-Moriya (DM) interaction α B 1 2 2 1 α∂ − – 5 – κ sin cos ) = − 3    n ( 1 = U denotes a one parameter family of models corresponding α ) − π ∇ 2 , [0 2 ∈ consists of a Zeeman term with the magnetic field : ferromagnetic phase ( α 0 : ferromagnetic phase ( U 0 . The three ferromagnetic phases of the potential : canted ferromagnetic phase ( A, B < A 2 2 A, B > ], and the rotated derivative is defined as 2 − 9 < , | 8 Figure 1 corresponds to the Néel type DM interaction. The subscript “bulk” indicates that B B > B < | 2 π For large values of the potential parameters 2. 3. 1. = These three phases exhaustfor the the possible ground homogeneous states ground of states. the potential The is phase included diagram in figure be neglected, the groundthree state types is of determined minima by of the the minimum general of potential the potential. There are where the parameter to different types of DMα term, for example we need to add avariational boundary principle), term as to we properly describe define subsequently. the DM interaction term (to apply the energy [ and an anisotropy term with the coefficient The term corresponding to the parameter where the potential JHEP02(2021)095 , 0 (2.8) (2.9) (2.4) (2.5) (2.6) (2.7) B < ] to the 31 , x. 30 2 ) is rewritten as d 4 | 2.6 w | (in the case of instead) A ) 2 3 ) in eq. ( is positive and the lower n 3 ( n x. V B U )) + 4 2 ¯ d w ) z (1 + , it is often more convenient to ∂ ~n 2 A . × 2 w . 3 ) = 1 α 2 2 n + | ) = 2 ) − instead of 3 3 . w ~n w ∇ | ) n n 2 z ( 3 ( 3 → − ∂ · n − in ( n V . ( 3 A, 3 ∞ iα with (1 + , the solvable line, since explicit skyrmion + V (1 n e ~n 1 A 1 + ~n 2 A = 2 in the following, since we can obtain identical n – 6 – R Im( Z 0 B B, n = κ = 2 κ ) = defined by the stereographic projection 3 ≥ with the top sign when w − determined by the homogeneous state. Explicitly, for n B w + 4 in the potential as ( 3 → − B ~n e 2 | V ] = B ± w ~n ) with the potential replaced by A, B = Ω[ |∇ 2 π 2.1 ∞ 4 2 ~n R to obtain a potential − Z A = ), we obtain a potential ) A A = 2 2.1 =2 B B bulk by just changing the sign of ] − 0 w [ is negative. E B B < is the asymptotic value of ∞ ~n In the case of derivative interactions like the DM term, particular care is needed to Instead of the constrained vector field If we tune the parameters Let us note that the model is invariant under the transformation Here the ferromagnetic phases, sign when sufficiently fast asymptotically, the surfaceobtain term obstructs the the field variational equation. principlethis used Hence to obstruction it and is allowinteraction, mandatory us to this to subtract is apply a accomplishedenergy boundary the functional by term variational ( adding to principle. the remove following In boundary the case term of [ the DM is because the magnetisationground vector of state the asymptotically. skyrmion configuration is required to goproperly to define the the energyneed functional. to When integrate varying by fields parts, to which obtain produces field equations, a we surface term. If the fields do not decay 2.2 Exact singleDefining skyrmion isolated solutions skyrmion states requires knowledge of the minima of the potential. This use the unconstrained complex field The energy functional in eq. ( We call thisconfigurations choice can of be parameters, constructed as shown in the next subsection. (and add a constant we should tune We consider the parameterresult region for JHEP02(2021)095 , ]. ]. 7 by 1 20 2 ) ), the S (2.15) (2.13) (2.14) (2.10) (2.11) (2.12) 2 S 2.6 as ( 2 2 π ), along the x. R 2 d 2.12  ) 3 plane) at n ( y . Some properties of V ) and ( − A x ) + 2.11 ~n we can regard , corresponding to × ]. α Q x. , one finds the exact solution by − while an anti-skyrmion has topo- → ∞ → ∞ 2 20 ϕ , ∇ r 1 r    ( ) and the potential in eq. ( . ) ) 10 · ~nd − . sin 2 γ γ )  r ∂ 2.9 ! 3 = + + e 2 κ Ar × = A κ Q ϕ ϕ − 2 ,  ~n 1 − x ~n κ A ( ∂ , 1 at infinity cos Θ · ϕ

κ = – 7 – ~n 1) π r , + cos Z 0 2 sin Θ sin ( sin Θ cos ( , r | = 4 π 1 ~n    (0 4 = E |∇ = 1 Θ = 2 arctan for a skyrmion configuration. It is worth repeating that → ] = 1 2 x 1 ~n tends to a constant as  ~n ~n [ − 2 ~n ), are discussed in refs. [ Q R (the magnetisation vector lies in the = Z 2.9 Q . As = 0 2 Ω = 3 S and profile function π . This is in contrast to the case of baby skyrmions as discussed in [ n 4 α − − A = +1 2 π =2 Q = B bulk γ E ) is crucial. This is an exact expression for the skyrmion energy as a function of ) shows that ] = 2.9 ~n [ 2.12 The skyrmion solution has energy The hedgehog solution is called a skyrmion. It gives a mapping from the base space The tuning of the potential allows one to obtain exact solutions of a hedgehog type [ Taking into account the boundary term in eq. ( We wish to stress that only by adding the boundary term, does the DM term become E to the target space along the solvable line. The analytic skyrmion solutions, eqs. ( 2 2 A κ important for our approacheq. to ( constructing the skyrmion lattice. The profile function in which is defined as the radius of a skyrmion. Here, when calculatingterm the ( energy of the skyrmion, takingsolvable line into enable account us to thesubtracting compute the boundary this boundary exact term energy leads expression. to the As skyrmion highlighted having in negative table energy which is very logical charge adding the point at infinity, thus we can define the degree This topological charge is in this context a skyrmion has topological charge R with phase In terms of the polarimposing coordinates a boundary condition well-defined in terms ofabout the the necessity variational of problem thethe for boundary boundary the term term are energy in discussed functional. eq. in ( appendix More details model of a chiral magnet that we consider is JHEP02(2021)095 . The (2.18) (2.19) (2.20) (2.21) (2.16) (2.17) | ] ~n [ Q | , of the form π 1 4 . x, T 2 ≥ − d 1) 4 and the anti-skyrmion | Q − ]. , 2 w ) we can complete the | 0 κ 2 , 30 A π π κ , = 4 4 ~n 20 = 1 A = (0 ) + 4 D A x, ¯ ~n 2 w = 2 z × d ∂ 2 ¯ = 2 B ~n BPS anti-skyrmion w ) 2 z 2 , | w B = ) , here 2 , w z iα w∂ coincides with the BPS solution with b ) | ~n 2 ( , ¯ z 2 − 2 | b 2 ¯ ∂ e = if iα κ w κw | − z − − i (1 + π iα e ¯ 2 z Im( π = 4 + 2 ¯ e ] where the energy is given by w 4 κ ¯ i z ¯ 7 2 z κ − A A (1 + i κ have energy given by their topological charge – 8 – , is a covariant (helical) derivative. In terms of 2 ~n, D = w∂ + 8 2 iα . Subtracting the boundary term is necessary for the 2 = z , z e 2 | ∂ w . This is in contrast to the usual story for soliton BPS skyrmion D ¯ z w ¯ ). The BPS case is x 2 w ) at = ∂ ) and find first order BPS equations 2 of the stereographic projection, the energy functional × z R = 1 ) = → ∞ d ), the BPS equations become ∂ Z  2.9 z w i ) as | ~n ) ( w 2 r 8 2.18 1 A π κ f − ~n 2.10 2.7 2 2 2.7 ∂ R = = − at π sigma model [ , for Z 4 1 × ~n ~n 1 Q  1 + (3) ~n ) with × π Skyrmion D 1 in eq. ( O → 4 in eq. ( ∂ α πQ ( w 2.21 − i 3 · w n A κe ~n ] ~n ] = 4 R =2 [ − w B bulk [ E ~n i E E ] = ] it was shown that for the BPS case ( ∂ ~n [ 20 = ~n πQ an arbitrary holomorphic function. Solutions of these BPS equations with the i . Full details of these BPS solutions are given in refs. [ . A table comparing the energy of the exact skyrmion solution and BPS anti-skyrmion with ) D 1 z ( − In ref. [ In terms of the complex field ] = 4 f = ~n [ boundary condition E solutions in the familiar exact Hedgehog solution inQ eq. ( and there are infinitely many BPS solutions, with topological charge for where the stereographic field where the centre of the skyrmion is denoted as square in the energy functional ( respectively. The single skyrmiongraphic exact complex solution field can be rewritten in terms of the stereo- and is constructed from eq.skyrmions ( to have negative energy. and topological charge can be expressed as Table 1 and without the boundary term, eq. ( JHEP02(2021)095 ) , we , the 2.22 2 (2.22) ) of a κ A 2.15 , the single ) and canted circle becomes → ∞ , below which an 2 = 0 = +1 A . The solvable line κ 3 3 0 n n = A . decreases down to 2 A > . 2 κ A  , and the spiral phase, for the ) and the radius ( 1 = reveals that the homogeneous = 2 2 κ − 2 A B 2.14 2 . On the other hand, the negative A κ 2  below A A < κ A covers most of the interior. In the same A > κ , indicating that the = 4 1 0  − 2 → A κ 2 – 9 – = ] showed that the inhomogeneous phases are the
A − κ 3 A and the skyrmion radius goes to zero. Conversely, n 14 1 ,  π π 4 π , the energy density is positive. As 13 4 , 2 κ 11 ) shows that a single skyrmion is a positive energy excited = . This leads to the skyrmion energy density being . If we decrease the potential strength parameter 2 ) along the solvable line i ) Sk 2.14 e 2.10 h κ/A → ∞ ( π A ) is approximately contained inside the circle of the skyrmion radius . In subsequent sections, we wish to shed more light on the nature of 2 higher than κ 2.14 and thus the ferromagnetic state is no longer the lowest energy solution A 2 in the limit A < κ with the area For values of As we have an exact expression for the energy ( Detailed numerical studies [ κ/A = +1 = 3 skyrmion energy increases towards the skyrmion radius diverges invery the large limit and thatlimit the the energy region of where agoes single to skyrmion zero. diverges. However, the average energy density ( We observe that the energy density is symmetric around skyrmion, we can computeskyrmion energy the ( energy densityr inside a skyrmion. We assume that the skyrmion lattice phase, at intermediateregion values much of below these inhomogeneous ground statesthan and numerically. their phase boundaries by analytic means, rather of the equations ofinhomogeneous motion. ground state This constructed suggests fromfirst that the observation there of negative the will energy precise skyrmions. betransition value point This a of cannot is the phase be our phase established transition precisely transitionof to by point the numerical a in analytical methods a new and single chiral shows skyrmion the magnet. solutions. power This expect that skyrmions willstate. start to Therefore appear weinhomogeneous expect as state defects with that many in there skyrmionsstate. the is is favored Repeating homogeneous a over this ferromagnetic the phase for emphasis, homogeneous boundarystates the ferromagnetic for analytic at skyrmion solutions become negative energy DM interaction becomes moresingle important skyrmion and energy can eq. ( resultstate in above a the negative ferromagneticenergy energy. ground of state The for the exact exactferromagnetic state single is skyrmion no solution longer for the lowest energy state. As We are now inthe a position energy to functional consider incoincides the precisely ( ground with state the offerromagnetic boundary the phases. chiral between Along magnet then the described ferromagnetic solvable by ( line, we obtain the ferromagnetic ground state 2.3 Instability of ferromagnetic phase JHEP02(2021)095 C (3.2) (3.1) ∈ ’s) get b w where the . The two = ~n z A by construct- 2 is used to su- 1 ~n − ] and consider the B > = 36 Q , centred at 6 B w . decreases the approximation  b ¯ iα A − − ¯ z  iκe A . B = w ], the magnetisation field B . Here, we work at the level of the complex + 2 36 A , − . The degree of the superposition is the sum of 6 w – 10 – B = ~n , w = Q iα w ¯ z − and , both solve the equations of motion. However, the A B A iκe w ~n , in contrast to the superposition in terms of = centred at the origin and another A = 1 and w . These figures show that as A 2 . A , in this case ~n w 2 A w B skyrmion configuration, which approaches a solution of the field . In this section we explain the approximation and use it to fit the ~n T 2  skyrmions. The centres of the skyrmions are defined to be the points . − 1 1 2 and A − κ fields, does not as the equations of motion are non-linear. As stated above the − = , A ) where the superposition is 1 0 ~n = w Q , − 0 2.7 Q  = = Q n We take a skyrmion The superposition in terms of the complex field has the advantage of automatically are included in figure depends on the value of scale which controls thepower validity law of decay the complicates matters dipole andlength approximation. it is scale. Along not as However, the obvious it solvable howfor to is line infinite extract still the a separation. known characteristic thatwith This the increasing means superposition separation. that approaches the adimensionless In accuracy true separation fact which of solution demonstrates we the that show approximation the below will accuracy that improve of it the approximation is also best to work in terms of a further apart. Some examplesA of these superpositionbecomes configurations less at reliable. different In valuesthe fact of skyrmion the diameter, key quantity this toskyrmion will fields consider be are is exponentially made the decaying, explicit the ratio decay below. of exponent the gives For a separation the characteristic to case length satisfying the constraint degree superposition superposition becomes a solution only inof the limit the of approximation infinite separation. improves as Hence the the accuracy centres of the skyrmions (the poles of the field in eq. ( We study the interaction between twoing magnetic a skyrmions of degree degree equations at infinite separations.perpose two In skyrmion refs. solutions, [ the degrees of where interaction energy as avarious values function of of the ratio of separation3.1 to the skyrmion Superpositions diameter of for skyrmions Although it is energetically more favorablepoint, to produce one skyrmions cannot below create the infinitely phaseit many transition skyrmions is because important of their toinhomogeneous interaction. study ground Therefore, state. the interactions Wecase between can of skyrmions use two the in method order from to refs. understand [ the 3 Interactions of magnetic skyrmions JHEP02(2021)095 1 β − (3.7) (3.4) (3.5) (3.6) (3.3) = Q , a iα ¯ − z as . On the left the A iκe 0) 2 hedgehog skyrmions , . This shows that as ζ, β 1 to be 1 2 = is also dimensionless. − κ − κ ] B there is a cancellation 1 w 10 w = [ , w (30 ζ. = 2 A Q int = 0 d w A E b , ! ]
, we show that the interaction B ¯ b. w , w B ζ . At [ iα 2 ∂ A . 2 − E 2 κ 2 ) e  2 w = 0 b ¯ | b − ¯ )] A κ ] b 2 w π − | − A Im ¯ z z = w [  2¯ ¯ z E + 4 (1 + ) + 16 | 4 iα − | . In appendix between two skyrmions is given in terms of the β , except ] , and on the right is − 2 | – 11 – z, β w b A 2 | ( | w b κ [ iα r/ | |

iκe 2 − 10 E sup 2 β 2 e | I = = A κ Z = ] = 2 w A | is a function of only two dimensionless ratios of the three w b [ . We define dimensionless coordinates | = ] ] = [( ) = | int ζ w w [ β [ as | E ( | int int β κ, A, b | for all values of E is dimensionless (a dimensionful constant is divided out from the E sup 2 2 I E − = , in the middle is Q 2 2 κ = ) A A . The magnetisation vector field for the superposition of two (2 / 2 κ We define the interaction energy of the pair of skyrmions decreases, at fixed separation, the interpretation of the superposition as two distinct skyrmions and The physical (dimensionful) separation dimensionless separation energy has the following simple structure as a function of two dimensionless parameters Our energy functional physical energy). With this convention, theFor interaction dimensional energy reasons, dimensionful parameters, configuration that is not a solution of the equations of motion. and regard it as the potential energy of the two-body force between the skyrmion pair. This has degree between the numerator and the denominator and we end up with plot is for A disappears and the approximation becomes worse. The superposition is then given by Figure 2 with one skyrmion centred at zero and the other skyrmion centred at JHEP02(2021)095 ,  ξ , as ) iθ (3.8) (3.9) r − . For . This e 2 π A κ Θ(  with the ) for the 4 of the DM w − is positive. w 3.6 is a real pa- iθ α = e w β , is to proceed E → depends only on ) = 0 sector is bounded ) we have ξ ( 2 A , sup 2 3.4 w 0 − I π. ≥ = 8 π Q ≥ − ). The integration is carried x ] + 8 2 ], again for 3.6 w d [ 2 34

E 2 2 ) and numerically integrate it using w . Using the superposition ) ] = iα 2 2 | . B 3.7 / | 2 w κe w β | [ using eq. ( | i κ 2 ] E − = w [ − (1 + w ] A are BPS skyrmions with energy ¯ – 12 – z int must be positive to respect the completing the A ∂ B E

w [ w int only linearly. The coefficient Z E E ) ) with − A and ] + 8 (2 ). Our approach proceeds by evaluating eq. ( A w , the interaction energy of the superposition π 2.16 / [ 2 2 w 8 3.2 E κ κ − region so that we can approximate the boundary as being at . We can assume without loss of generality that = ]. Another approach used in [ 6 B ] = ] = A w 10 w 36 [ w , [ + in eq. ( × int = 2 35 E A 2 B E w B w in [ by = which we consider, we find that the interaction energy between the exact 6 , we obtain w . Now from the definition of the interaction energy in ( 2 and = 0 A 10 π − . We compute the integrand of eq. ( 8 A A | × − w = β 2 | Q ] can be applied to eq. ( 20 It was not known if the interaction energy isFor positive general external for magnetic fields other and anisotropies values there are of no known exact solutions means that the interactionaration, energy just depends onMathematica, the to modulus find of an theout expression dimensionless in for sep- a infinity compared to the skyrmion separation. After repeating the integration at many 3.3 Computation ofWe the are interaction now energy readysolutions to compute the interactionsuperposition energy between two exactrameter. single This is skyrmion because the invariance of the energy under rotations, so a different approachthis is is needed to to consideris study the done the asymptotic for interaction form energy. ofnumerically. the One equation Both way for of to thedecreases these approach hedgehog exponentially approaches for profile, find increasing separation. that the interaction energy is repulsive and for asymptotically large separations. the range of skyrmion solutions is always positive. where we have used thatmeans both that the interactionsquare energy, argument. We alsoinfinite separation. know Combining that these the facts, interaction we find energy that must the vanish interaction is in repulsive the at least limit of The second term isother zero configuration. when This the meansbelow that BPS by the equations energy are in satisfied the and degree is positive for every This is demonstratedref. in [ the following way.degree The completing the square argument from interaction, and depends on the magnitude of the dimensionless separation, 3.2 Qualitative analysisAt of the interactions BPS point We observe that the interaction energy is independent of the angle parameter JHEP02(2021)095 is A 2 π (3.11) (3.10) 6.0 )+16 is plotted | decreases. β ) | , the inter- π ( 2 2 A κ and the red fit κ 5.5 sup 2 π I ) + 16 | β | + 16 ( 5.0 sup 2 sup 2 I Log|β| I . There is good agreement 4.5 , the phase transition point . On the left log ( 2 3 κ 500 = . decreases towards becomes smaller as < 4.0 | | A . β 735 A β | . | 1 | 735 < 1592 . β 1 | 3.5 | 30 1592

β 1 0 . As

-1 -2 -3 | '

2

) π + ( 16

Log 2 I Sup A κ for 2 π A decreases, and it is likely that we are over- we fit the interaction energy as a function of κ 2 π , while on the right | is plotted in figure A 500 – 13 – ' β | ) marginally over estimating the strength of the π ] ) + 16 )+16 | | w = 500 [ β β | | | 3.11 ( ( int β | ) + 16 400 E | sup 2 sup 2 using a log-log plot and find: β I | I | ( the distinction between the two single skyrmions becomes and β | A sup 2 300 I = 30 |β| | β | 200 . The data points are the numerically computed values of | . 100 β | 735 . . These are fitted plots of 1 | 1592 log β |

0 3 2 1 4

2 π + 16

I The interaction gets stronger and the skyrmion energy becomes more negative as The general trends that we observe are that skyrmions are small, weakly interacting, The power law fit for Sup Therefore our results getestimating the less repulsive reliable interaction as where energy. for smaller This values canless of be pronounced. observed At directlyothers short from shape distances, figure and the a skyrmions more will sophisticated have method a is significant needed to effect account on for each this. skyrmion interaction gets stronger and the skyrmionswith increase the in size energy and of decrease an inbetween energy, individual the skyrmion ferromagnetic vanishing and at skyrmion lattice phases. decreases. On the other hand, the relative separation between the fit and thethere numerical computation is for a larger separations. slightinteraction. At deviation small with separations eq. ( and have positive energy for large values of The interaction energy is then line is separations, between the dimensionless separation Figure 3 plotted against the dimensionless separation against JHEP02(2021)095 (4.1) (4.2) defined , ` ! 1 735 . C 1 2 κ A` 2 is the constant given + 3 735 . 1 1592 Sky 4 E ' κ/A

1 2 ` C = 2 2 3 r √ 2 . | 2 . The average energy density is then 4 β  | ), and 4 κ A 2 =  2.14 . A κ | | 2 b β | | ) = – 14 – = int in the calculation of the two-body force between | E r b ) for | 2 | b | + 3 4.1 = ` Sky , as shown in figure E 2 ( ) 2 ` κ` A 2 2 3 ( √ 3 2 2 √  ). We find it convenient to use a dimensionless separation = . The unit cell of the triangular lattice is the red parallelogram. κ A 2 2 3.2 | to the diameter of a single skyrmion  b | | b 3 | 2 √ ) = ` is the skyrmion energy given in eq. ( ( Figure 4 ) when substituting eq. ( , the homogeneous ferromagnetic state becomes unstable due to the emergence ] indicated that a triangular lattice configuration is the ground state for the i 2 . In light of this numerical work we construct a simple model for a triangular Sky 11 3.11 A E 2 Lattice Assuming that the two-body force between skyrmions is the dominant interaction, we A < κ e h where in eq. ( Using the interactionapproximate energies energy computed density for in athe triangular the energy skyrmion previous per lattice. section, unit Ourand cell. approach we has is The area to now unit compute compute cellgiven the by contains one skyrmion, with 6 nearest neighbours, lattice spacing to beskyrmions the in separation eq. ( as the ratio of skyrmions, was also found numericallyB in another < parameter region withlattice larger of anisotropy, skyrmions. Usingfind this the lattice model, spacing we which compute minimises the this. average energyestimate per the unit average cell energy and density of the triangular lattice state. Let us identify the is expected to lead toor a square lattice of lattice skyrmions. configurationin Depending will on have ref. symmetries, the [ either lowest aparameter energy. triangular range An where extensive numerical thelattices study exact of skyrmion vortices solutions in occur, superconductors as and in superfluids. the case A of cubic Abrikosov lattice of merons, half 4.1 Average energy densityFor of skyrmion latticeof state negative energy single skyrmionresults solutions. in The a negative preference energyinteraction to of prevents produce individual the as skyrmions skyrmions many from skyrmions becoming as too possible. densely However, packed. the repulsive This competition 4 Skyrmion lattice states JHEP02(2021)095 . 1 ` ≥ (4.3) 125 2 A κ 100 with is negative at its , and at the phase i 75 1 2 min whose minimum is for ` it reaches a minimum ℓ 2 = κ 2 = A Lattice 2 κ A ` κ e = h 50 A ). On the left is the critically , respectively. These figures 7 4.2 , δ 1 25 leads to . `   2  A κ configuration, may be lower. Namely 1 1 2 C 0 −

− and figure

) 3 0.0006 0.0004 0.0002 Lattice 2 ) ℓ >( < e A δ 6 κ and the dimensionless lattice spacing at the configuration, we naturally expect that the = for the BPS case,  ) . In the BPS case and on the right is 2 ` Q π − (2 + 11 min 125 ) will have different behaviour as a function of – 15 – = 1 16 ` = ( ' 2 − becomes small. Hence our calculation of the inter- i A in figure 4.2 κ ` Q |   ) with respect to 2 β A κ | ' 100 , 4.2 Lattice 2 , eq. ( e A κ 2 h min A κ ` as a function of 75 i is positive and reached for asymptotically separated skyrmions. ℓ i Lattice 50 e with its minimum at h Lattice minimising eq. ( e 2 2 h decreases the dimensionless lattice separation at the minimum decreases. κ 1 . , are plotted against 2 before increasing towards zero. = A < κ 25 . Thirdly for small 735 3 2 A . min . A κ ` we plot = 1 . These are plots of the energy per unit cell from eq. ( 5 = 1 = 0 0 2 δ A

2

κ 0.003 0.002 0.001

A -0.002 -0.003 -0.001 Lattice κ ) ℓ >( < e Let us make a few observations. Firstly we observe that a finite lattice spacing is The average energy density When Depending on the value of solutions as a variational Ansatztrue for total a energy at ait fixed is separation, quite for a likelythe that repulsive our interaction. approximate Therefore resultwill we be for expect smaller the that than the interaction our skyrmion energy estimate separation is for in overestimating small the values lattice of achieved by the balancepositive between energy the due to negativeanalytic the energy understanding repulsive is of most interaction constituent reliable betweenpoint quantitatively skyrmions skyrmions. for and regions Secondly, near the this theaction phase semi- energy transition becomes less reliable. Since we are using a superposition of single skyrmion minimum, show that as While at first the latticenear energy density decreases with decreasing where the minimum of This is because the ferromagneticenergy state excitations is above the it. ground state and skyrmions arise as positive In figure transition to the ferromagneticminimum phase, and there the triangular skyrmion lattice is the ground state. While for Figure 5 coupled case infinitely separated skyrmions. JHEP02(2021)095 ) 4.3 , we from . As , the (4.4) 2 2 κ A A κ κ 2 min ` from eq. ( ) against min ` ), at the minimum 4.3 4.2 . will not attain the value , with the average energy ~n 1.0 min 0 ` from eq. ( → 0.8 min ` 0.8 A as 0.6 0.6 κ 2 A → ∞ A κ 2 c A κ would have skyrmions separated by κ A 0.4 the lattice spacing and the skyrmion diameter = 2 = – 16 – | 0.4 2 = 1 A b κ | r c , compared to the DM interaction parameter 0.2 A 0.2 0.0

and radius

0.000

5 0

-0.001 -0.002 -0.003 -0.004 25 20 15 10 Lattice min min ) ℓ >( < ℓ e . On the other hand, the exact single skyrmion solution has energy 2 A κ → −∞ ) we see that for small 2 A also decreases. The blue curve is a plot of the approximate value of κ 7 ] that spiral states have lower energy than the lattice. Thus we expect that . The blue line is found by substituting the approximate value of inside the radius tending to a finite value. − 11 min A i ` ) and the red points are found numerically. (1 Sk π . Here we plot the dimensionless lattice separation . This figure shows how the average energy density of the lattice, ( e 4.2 h a positive constant. The case of ) and the red points are the numerically computed values of = 4 c From figure 4.3 decreases Sky kind, we consider the case of skyrmions separated by a multiple of the skyrmion diameter, with skyrmion diameter, which means that the magnetisation vector E density become the same orderlattice of and magnitude. spiral phases This we suggests need a that better near model the of transition the lattice. between the As a candidate model of this For smaller values of theexpect parameter from [ a phase transition between thein skyrmion the lattice preceding and section, spiral ourfor states estimate smaller occurs. values of of interaction As energy we is have likely seen to be an overestimate Figure 7 A eq. ( 4.2 Transition to the spiral state Figure 6 changes with into eq. ( JHEP02(2021)095 | , ]. b | 39 that = 0 (4.5) (4.6) (4.7) ) has A A 4.5 , since we 2 ] for A κ 10 ). The transition will is much larger than the | b . We are also assuming | C.5 and the lattice separation = 2 r c 2 , both the skyrmion diameter and lattice A 2    . 3 2 ) ) 2 A κ 2 κ + κx κx the lattice spacing, 2 is obtained by rotating and/or translating − 2 0 κ A = sin ( − cos ( i = 0 − = – 17 – B i    spiral = , decreases. The scale in the figure on the right has been e = | h r b 2 | spiral A ~n and will in fact focus on ) was motivated by comparing the energy functional e h ) there is an exact spiral solution = 1 ` 4.5 . While it is not a solution of the equations of motion it = 0 c > C B . On the right for lower r = 2 plane with the simultaneous rotation of the magnetization vector . On the left for larger A 2 A y κ - x (at ]. This scenario seems to be more realistic for small values of . A rough estimate of where the transition between the skyrmion lattice . This type of triangular lattice made from nearly touching skyrmions was 17 8 ) = 0 = 0 3 n ( κ~n . A schematic of showing how the skyrmion diameter V a spiral state like eq. ( ) to an energy functional whose ground states are Beltrami fields, solutions to ). When the potential is non-zero the spiral configuration given by eq. ( + 2.1 ~n 4.5 Using this alternative model of a skyrmion lattice we can estimate the value of 6= 0 between the skyrmions. A situation like this is called a hexagonal meron lattice in [ . This has an identical energy density so we choose to work with the simplest case, 3 B eq. ( ∇ × and the spiral takes placea can single be skyrmion. found by The comparing skyrmion this average to energy the is average given energy by density eq. of ( eq. ( average energy which is shown inprovides an appendix approximation for a spiral state when the potential is small. In [ The most general spiral solutionthis at solution in the with average energy considered in [ are overestimating the interaction energy in that region. the transition between the latticethat for phase to the spiral phase occurs at. To do this we note e As such we arethat interested in the repulsive skyrmionfrom interaction merging, can such that beright they neglected, in form figure except a to lattice. keep This the leads skyrmions to the situation depicted on the Figure 8 change with decreasing diameter of the skyrmion, separation increase but their ratio, exaggerated for effect. ~n JHEP02(2021)095 , 1 (4.8) (4.9)  ]. We 2 , where (taking 11 A κ A κ A . To achieve = 2 = 4 A | B ]. For b | , where the exact = 2 20 B = 1 2 A κ , the exact skyrmion solution , 1 2 κ < 22 2 . A . κ 0 2 κ ' 17  . 0 17 √ 4 as the ground state. This ferromagnetic phase – 18 – i ' − − 20 ]. As the skyrmion energy is known analytically the is the magnetic field parameter in the potential, so = 1  spiral e 3 2 B 14 h , 16 κ n until we reach the critical point at 13 = , 2 for the explicit details of this computation. At this value A κ A 11 C , is found exactly without the ambiguity of numerical simulations. 2 κ = . Our results have also demonstrated that in the skyrmion lattice phase using the dipole approximation. This is further evidence that our dipole A )) this gives the phase transition at matches the location of the phase transition found numerically in [ 6 = 1 4.4 A 2 A κ in eq. ( decreases the ratio of lattice separation to the skyrmion diameter decreases. This sug- We have studied the properties of this skyrmion lattice state by working out the inter- 2 A κ is the anisotorpy parameter and = 2 predict that the phase transitionhappen from a for skyrmion lattice stateas to a ferromagnetic state will gests that near the boundary betweenimately the computed lattice by and taking spiral the phases lattice the density lattice to can be be the approx- energy density of a single skyrmion. this we used the dipolesolutions approximation centred for with the a finite superpositiondependent separation. of of two, and the exact, have angle found single parameter that skyrmion other of the types the interaction of DM energy skyrmion. interaction is which Exact in- distinguishesand expressions after Bloch, for numerically the Néel integrating interaction or these energy density awell were power separated obtained law fit skyrmions. is Using found for the the computed interaction interaction energy of energies, we have been able to is found numerically in refs.transition [ point, This is only possible on the solvable line whereaction exact between skyrmion exact solutions skyrmion are configurations known. along the solvable line continues for smaller values of skyrmion solution has zerohas energy. negative energy. Below this The presence point means of that a the negative homogeneous energy ferromagnetic solutionsuggests state to is that the no the equations longer of true the motion lowest ground energy state state. becomes This an inhomogeneous skyrmion lattice state as the phase structure of a chiralA magnet. We have focused onthat the solvable we line can makewhere use the of DM interaction the canhomogeneous exact be ferromagnetic skyrmion neglected state configurations compared to from the ref. potential, [ we have found the approximation underestimates the energyaction density energy. because it is over estimating the5 inter- Conclusion In this paper, we have studied the interaction between two well-separated skyrmions and of the coupling the spiral state has energy density This energy density is muchlattice lower in than figure the energy density that we found for the skyrmion c this value of refer the reader to appendix occur when these two energy densities are equal. When the separation is JHEP02(2021)095 (A.1)  , a α in the bulk with the  ∂U ∂n  bulk α b ] + n ~n α b through the variation of the [ n aib E i ∇ κ bulk ] ~n − aib [ α a κ n δE i ∇ + 2  α a α a n 2 δn  ] that we have used are minimisers of the static i −∇ – 19 –  ∇ 20 α a + δn  x 2 d Z at which the single skyrmion energy becomes negative away = ). To obtain the equation of motion by the variational principle, B ]. Extension of the present work to three dimensions remains an a α 2.1 40 bulk ] δn and ~n [ A δE This work is supported by the Ministry of Education, Culture, Sports, Science, and We have studied skyrmions in two dimensions, applicable to a thin film. In three spa- We need to emphasise that our work deals only with static configurations, since the We have emphasized the need for the boundary term in the energy functional to make magnetization vector In this appendix, wetional explain is why mandatory. the inclusion LetDM of interaction us the in consider eq. boundary the ( termwe energy consider in functional a the variation energy of the func- bulk energy functional the Promotion of Science (JSPS)Number Grant-in-Aid (16H03984 for (M.N.), Scientific 18H01217 Researchported (KAKENHI) (M.N. in Grant and part by N.S.)). aMaterials The Grant-in-Aid Science” for (KAKENHI work Scientific Grant of Research No. M.N. on 15H05855) is Innovative from Areas also MEXT “Topological of sup- A Japan. DM interaction and boundary terms skyrmions and the importance of thediscussions boundary term, about and interacting Sven solitons. Bjarke Gudnason for general Technology(MEXT)-Supported Program for the StrategicUniversities Research Foundation “Topological at Science” Private (Grant No. S1511006) and by the Japan Society for open problem. Acknowledgments C.R. thanks Bernd Schroers and Bruno Barton-Singer for useful discussions about magnetic find the values of from the solvable line. tial dimensions, skyrmions are linesalong for a which skyrmion one line can [ study collective modes propagating behavior of the magnetisation vector. exact skyrmion configurations from ref.energy [ functional. Understanding theground effect state of is timeskyrmion an evolution configurations of open excited and problem, states their above which superposition. the requires Another an interesting open understanding problem of is the to dynamics of net along the solvable line.between In the particular homogeneous we ferromagnetic determineand the phase estimated location and the of transition the the between phase skyrmion the transition lattice skyrmion phase lattice precisely, phasethe and DM interaction the term spiral well phase. defined, and have clarified its close relation to the asymptotic This allows us to gain a semi-analytic understanding of the phase diagram of a chiral mag- JHEP02(2021)095 , is ∞ . It ~n Ω ) = 0 (A.6) (A.7) (A.8) (A.2) (A.3) (A.4) (A.5) ∞ of the ≡ a, n 1) bound , tangent to = 0 α, a bound i , n ~e b α, . We can take i n , ∇ ω, α a ) solution in the , bound ( = (0 x. 1 n 2 α, a  2 bound α  − n d aib , dx ~n α, a   ) bound ) using an identity in = n ) ∧ 3 ~n bound α, a . When we evaluate the 1 ) instead of the generic n bound Q n α, b ( A.4 × . α, b A n dx α U  n → − and the vector aib bound Z ) aib = 2 ∇ α, ) + κ − ( n κ ~n · bound B − components. The vorticity α, b bound − × bound as ∞ ) = n 2 α, a α, b α α 1 α α a n n − n , n n i n 2 bound 1 ). , ∇ ∂ bound ∇ x κ ~n n α a ( α, a aib ( 2 limit. The volume α, a · n n α a − d i κ n 2.10 ) n i 2 α 2  3 ∇ R e i n ∇  Z 1  ∇ α a − ∂ → ∞ π – 20 – x α a ( 1 4 2 ~n κ δn n ( d R 2   = i i κ Z is given by dx ∇ ∇ + ∧ x x = for the solvable line ( B . However, the circumference of the boundary increases 2 2 2 x ω Ω 1 | ) d d 2 ≤ π ~n 3 d 4 dx n Z Z 2 A ( |∇ R → ∞ 2 − Z − − 1 2 Z V boundary  − R . Therefore we need to consider only the second term. Thus the . Since the minimum of the potential is at π and take the = = 2 1 E bulk 4 R B = R E Z = 1 ≤ 2 = ) Ω = ] = A 2 ~n boundary boundary [ E boundary E bound δE E a α, n ( stands for the vorticity density in the ] ω , we obtain our energy functional in eq. ( ) 20 In order to study the contribution from the boundary energy functional in the case For definiteness, let us consider the case of single skyrmion ( 3 n ( of boundary at infinitycircle precisely, with we the need radius parallerogram to formed regularize by the thethe three boundary boundary unit by vanishes vectors as taking a large By choosing the potential U defined as in order to makeof variational the principle skyrmion well-defined. solution for Thus we find that the correct energy is convenient to rewriteref. the [ boundary energy functional in eq. ( where parameter region we should impose a boundary condition fixing the boundary value as where we denote the boundarycontribution value from of the the first variable term,because we of note that boundary energy functional is finally given by where the booundry value ofthe the following magnetization boundary vector energy is denoted functional as as a solution of the above requirement last term). We can(surface obtain term the in equation the ofadding variation motion a of if boundary the and term total only to energy) if the is energy the absent. functional total whose We derivative can variation term should achieve satisfy this goal by where we have performed a partial integration resulting in the total derivative term (the JHEP02(2021)095 ] in 10 ) C . In 1 (A.9) 0 (A.11) (A.12) (A.13) (A.10) − ). = , the fields 3 B > n A . A.11 = 2 ) gives the decay -axis, ) and keeping only B z , sin (2Θ) A.11 Ar A , A.10 2 α ϕ . It has been noted [ − − . B the boundary condition on /R √ 1 − solves equation ( e , sin Θ ∼ Θ = 0 r r 1 → ∞ Rdϕ κ n 1    B α ϕ  √ ) ) r π ∼ ) n 2 γ γ + ) 0 A ' : r ). Solving eq. ( Z 2 + + Θ depends on the fall off conditions on  0 2 − ϕ ϕ r − /R Θ( ) 1 Ar 2.11 = B sin 2 δ~n , where the boundary condition is given by ) sin ( α i cos ( 0 − × γ + (    can be rewritten into an integral along a circle ~n . B ) 2 κ n + ( 1 i . In the limit r r – 21 – in eq. ( √ A > α   2 dx ) . Considering the limit of ( ∇ · 1 r → ∞ C − A, B ≥ + Θ( sin ( I K r Θ( 3 boundary r Θ ), the fields and their fluctuations fall off exponentially κ B − dr e 1 d ) = 0 2 E √ r r 1 = = − 2.10 ∼ Θ( ~n + ) is the modified Bessel’s equation and thus ) r 2 2 Θ 2 r →∞ dr Θ( r 2 d A.11 centered at the location of the skyrmion (defined by boundary sin (2Θ) E lim R + case of eq. ( for large values of ]. we have Θ dr d 1) 31 , Θ = 0 r 1 . There are two cases for this equation related to different forms of Bessel’s 0 , equation ( , it is straight forward to check that , ~n − A ]: A A 2 = . As a total derivative, plane. Therefore the boundary energy functional contribute a finite result if the is the tangential component of the magnetization vector at the large circle 41 . This is discussed from the more general gauged sigma model point of view in = (0 2 2 = 2 α ϕ Θ 2 ], the ]. In the familiar case of a skyrmion energy including just a Zeeman term studied in δ~n n ∞ , x dr d decaying exponentially fast as B B > 1 → ∞ ~n 10 31 To showcase the explicit decay properties of the hedgehog configurations consider the x 2. 1. R with a large radius and → retaining terms up toproperties first of order in equation [ Asymptotically the hedgehog magnetisation vector becomes the profile function is terms linear in We focus on theparticular, we case consider of the case a of magnetic field aligned with the positive fast and the boundaryand term fluctuations is fall zero. off morebe However, slowly along well and defined the the [ solvable boundary line, term is not zeroradial and equation may of not motion even magnetization vector decays onlythat as the a value power of of the boundary term ref. [ ref. [ C where the as ~n ~n JHEP02(2021)095 , ). ). A 2 3.5 (B.2) (B.3) (B.4) (B.1) 2.16 ), these B > ), with the 2.18 in eq. ( , ib B.1 ! −
ze ¯ w A ζ κ 2 ∂ 2 2 ) = 2 w | ). , β w ) (the canted ferromagnetic | ζ. | . 2 Im | β A d | 2.14 β ( 2 for the superposition are only | 2 ! + 4 (1 + 2 , and I
), since the boundary condition 4 I ¯ | iα w , ! ζ 1 − w 2 | A ∂ I 2 κ A.11 2 2 < B < ze )

2 w A 0 κ

| 2 A ζ, w π, π, | 2 8 = ) + Im d + 4 | − ) and ( 2 ζ β 2 ) | 2 ) 2 + 4 (1 + ( | 2 | 2 | 1 | 4 ¯ ) = ) = 8 w | w ¯ I – 22 – A.10 w | β β ζ w | ζ w ( ( ∂ | | 2 1 ∂ | ] + 8 I I

8 w (1 + [ (1 + Z under the change of coordinates, it is , the magnetisation vector field decays slowly and the 2 Z ζ πQ  A 2 d A can be rewritten as ) to the case of a single skyrmion, such as in eq. ( κ . ) = ) = 2 2 2 | | ] = 2
1) ] = 4 β β  w | | , A B.2 [ κ w ( ( 2 0 B [ e + 1 2 , I I E Q = πρ x 6= (0 2 d ∞ ~n ] = 4 w [ e (the phase boundary between the canted and polarised ferromagnetic phases), the topological charge density. The energy is the integral of eq. ( is found to be linear. This structure is valid for any field configurations including ) A Q A ρ One should note that the functional dependence on the dimensionlessApplying parameter the formula ( This means that for (2 = 2 / 2 integrals can be evaluated explicitly to give which is consistent with the expression for the energy in eq. ( κ our superposition of two skyrmions.functions The of two the integrals magnitude of the dimensionless separation, where the two dimensionless integrals are defined as with measure scaled as In this appendix, we calculateless an parameters. expression for The energy the density skymionThis with energy the can in boundary terms be term of removed written dimension- change is in given coordinates in terms to eq. of ( theThen, dimensionless two the ones dimensionless energy parameters density in the following way. We phase), we have todifferent analyze differential equations the from asymptoticis eqs. behaviour now ( of different, the magnetization vector using B Expression for the energy density boundary term ishowever, needed the to magnetisation make vector fieldin the decays the variational exponentially energy problem fast isB well and the zero. defined. boundary term Therefore For from the the viewpoint boundary of the term variational is principle. mandatory For for our solvable line JHEP02(2021)095 is ] w ) the (B.9) (B.5) (B.6) (B.7) (B.8) (C.3) (C.1) (C.2) [ is Q 4.5 ! = 0 A dx ) = κx B ) the degree cos(2 2 . A and using the definition 2 L κ ) 2 L 2 2.17 − β ] B ( Z . π w 2 1 L 2 + κ sup 1 2 1 ζ. . I A − 2 2 π w A d is not zero then for eq. ( ) + 16 + 1 π. κ 8 2 = 2 β | ] = ( A − ¯ dx 2 w π ) dx  ) w ζ    2 3 2 = sup 2 ∂ ) ) | ] = 16 κx n . If I ζ w 2 w | κx κx spiral − | [ + d ) + 16 e 2 0 3 2 cos( = 0 | β ] + [ 2 L ) 2 ( n L πQ 2 i L π 2 w sin ( 2 cos ( 2 L 2 − | 8 (1 + 3 | ζ − Z n ¯ 16 ∂ − w − sup – 23 – w Z 2 − | | h ζ 1 I L    1 − 1 L ∂ | Z  ) 2 ) = = = β 2 1 L π (1 + β 2 L ( ] = [ − ( ~n − =0 w Z 3 2 Z [ of two single skyrmions is an anti-holomorphic function , and thus so that B . sup 1 ] = sup i 1 1

1 L π I int B A π I w 2 κ [ ) to the superposition 2 3 A A E − 6= 0 w are the dimensionless integrals for the superposition field ), we can write the interaction energy as Q spiral β ) = = = = + ] = [ e B.2 β ] = h 3.4 2 w A ( L ) [ w 3 [ w ) to int n sup Q 2 = E I − B.5 w (1 . The key observation for this is that in eq. ( ) 2 L and , as long as 2 L β 2 − ( ) . Z − β B ( 1 L sup 1 w I ] = A is anti-holomorphic, such as for a superposition of skyrmions, it becomes sup + 1 w I [ w A Q w For purely anti-holomorphic functions we can take this a step further and explicitly Applying the formula ( = The period is definedboundary as conditions imply that which has average energy C Spiral state In this appendix, we discuss spiral states. The exact spiral solution for This simplifies eq. ( When The superposition with compute written as of the interaction energy ( where w JHEP02(2021)095 , (C.9) (C.4) (C.5) (C.6) (C.7) (C.8) (C.10) 465 Nature , is κ A (1989) 178. c = 95 . . r we are in the ferromagnetic   4 4 2 c c 2 , κ , κ + 9 + 9 A = 2 . = 0 22 3 . 2 2 . c c A 0  4 , A 8 2 + 1 κ 3 A ' 2 2 − 2 c κ +  2 + 2 − A κ 17 2 64 + 16 64 + 16 κ Zh. Eksp. Teor. Fiz. A , for which we find  √ = p p , 2 − 4 κ A 2 κ A  , as above − ± c 2 = – 24 – 4 1 2 − c 2 2 κ i = 2 c c − = 20 8 r ≤ 2  i Thermodynamically stable vortices in magnetically , that is our lattice consists of a single skyrmion in a A κ 2 spiral 8 + 3 Sk A e 16 κ 8 + 3 8 + 3  e h ]. h   = 2 − matches the value that can be read off the phase diagram . 2 A = ), which permits any use, distribution and reproduction in 2 2 2 c c 11 4 A 16 16 κ κ Topological properties and dynamics of magnetic skyrmions A A = = A A (2013) 899 8 CC-BY 4.0 Real-space observation of a two-dimensional skyrmion crystal results in This article is distributed under the terms of the Creative Commons at which we expect a transition between the ferromagnetic phase and the A A . is positive and we know that Nature Nanotech. ordered crystals. the mixed state of magnets (2010) 901 The energy density of a single skyrmion in a disc of radius This results in A. Bogdanov and D.A. Yablonskii, X.Z. Yu et al., N. Nagaosa and Y. Tokura, c [2] [3] [1] Attribution License ( any medium, provided the original author(s) and source areReferences credited. as the value of skyrmion lattice. This value of that was numerically found in [ Open Access. An interesting choice is tounit take cell which is the disc with radius As phase, we take the negative sign and get solving this for Setting this equal to the spiral energy density we find which leads to the following quadratic equation for JHEP02(2021)095 ]. . , , , ] , SPIRE ] IN (2013) . Theory of . Phys. Rev. 341 (2018) 184303 , 97 (1976) 449 [ ]. (2009) 915 Science ]. ]. arXiv:1406.1422 , 24 [ arXiv:1006.3973 (2014) 20140394 323 [ ]. SPIRE SPIRE . SPIRE IN 470 IN IN ][ Phys. Rev. B ][ ][ SPIRE , Science Skyrmion Lattice in IN , [ Multi-solitons in a two-dimensional (2015) 224407 (2016) 60 Skyrmions in two-dimensional chiral , Cambridge monographs on (2010) 094429 91 ]. 56 Sov. J. Nucl. Phys. . 82 Spontaneous skyrmion ground states in , Magnetic Skyrmions at Critical Coupling (1975) 760 SPIRE Proc. Roy. Soc. A hep-th/9406160 IN , 35 arXiv:1812.07268 arXiv:1402.7082 [ – 25 – . [ [ Skyrmion fractionalization and merons in chiral . (1994) 255 An Exact Classical Solution for the ’t Hooft Monopole Phys. Rev. B . . Edge instabilities and skyrmion creation in magnetic , Metastable States of Two-Dimensional Isotropic Phys. Rev. B Chapter one — theory of monoaxial chiral helimagnet 138 Topological solitons , (1995) 165 (1975) 245 [ (2006) 797 Thermodynamically stable magnetic vortex states in magnetic (2020) 2259 Compactness results for static and dynamic chiral skyrmions near 65 22 (2014) 031045 442 (1958) 241 Phys. Rev. Lett. (2016) 065006 ]. 4 375 , 4 (2018) 132406 18 Skyrmion lattice in a chiral magnet Calc. Var. Part. Differ. Equ. Writing and Deleting Single Magnetic Skyrmions Skyrmions in thin films with easy-plane magnetocrystalline anisotropy , , vol. 66, pp. 1–130, Academic Press (2015). 108 A thermodynamic theory of ‘weak’ ferromagnetism of antiferromagnetics Nature Stability of Classical Solutions SPIRE , IN Z. Phys. C JETP Lett. [ , , Chiral skyrmions in the plane Anisotropic Superexchange Interaction and Weak Ferromagnetism Phys. Rev. X J. Magn. Magn. Mater. ]. ]. , , New J. Phys. , (1960) 91 . SPIRE SPIRE IN IN the conformal limit and the Julia-Zee Dyon magnetoelastic resonance in a monoaxial chiral helimagnet Commun. Math. Phys. Ferromagnets Two-Dimensional Chiral Magnet [ Solid State Physics crystals magnetic metals magnets Appl. Phys. Lett. layers magnets with easy-plane anisotropy [ mathematical physics, Cambridge University Press, Cambridge (2004). J. Phys. Chem. Solids 120 636 Skyrme model I. Dzyaloshinsky, T. Moriya, N. Romming et al., B.M.A.G. Piette, B.J. Schroers and W.J. Zakrzewski, N.S. Manton and P.M. Sutcliffe, S. Muhlbauer et al., L. Döring and C. Melcher. E.B. Bogomolny, M.K. Prasad and C.M. Sommerfield, B. Barton-Singer, C. Ross and B.J. Schroers, A.M. Polyakov and A.A. Belavin, J. Kishine and A.S. Ovchinnikov, A.A. Tereshchenko, A.S. Ovchinnikov, I. Proskurin, E.V. Sinitsyn and J. Kishine, A. Bogdanov and A. Hubert. U.K. Rößler, A.N. Bogdanov and C. Pfleiderer, J.H. Han, J. Zang, Z. Yang, J.-H. Park and N. Nagaosa, M. Vousden et al., J. Müller, A. Rosch and M. Garst, C. Melcher, S.-Z. Lin, A. Saxena and C.D. Batista, S. Banerjee, J. Rowland, O. Erten and M. Randeria, [8] [9] [5] [6] [7] [4] [22] [23] [24] [20] [21] [18] [19] [15] [16] [17] [13] [14] [10] [11] [12] JHEP02(2021)095 , ] . (2019) (1995) 15 62 ]. ]. Nonlinear , ]. SPIRE (2013) 214419 ]. IN ][ JETP Lett. SPIRE 87 Nature Phys. , IN arXiv:2001.10038 ]. , SPIRE ][ ][ IN Domain walls that do not get ]. ][ DOI SPIRE arXiv:2001.00273 [ Instantons in Chiral Magnets IN . ][ 11th International Symposium on SPIRE Solitons in the Higgs phase: The Phys. Rev. B IN Particle model for skyrmions in , . hep-th/0602170 ][ , in Nonlinearity and Topology [ ]. , Cambridge Monographs on ]. BPS soliton-impurity models and Decays and Quenching of the Weak Coupling Towards an Effective Theory of Skyrmion Skyrmion-Skyrmion Interaction in a arXiv:1403.4031 Magnetic skyrmion interactions in the [ (2020) 415803 (2020) 134422 , Dover Books on Mathematics, Dover ββ SPIRE arXiv:1910.13907 SPIRE IN 32 [ IN 101 – 26 – [ and (2017) 014423 (2006) R315 arXiv:1901.04501 β [ . 96 39 arXiv:1907.02062 , (2019) [ (2014) 025010 [ (2019) 30 Asymmetric isolated skyrmions in polar magnets with 7 90 Nonrelativistic Nambu-Goldstone modes propagating along a Phys. Rev. B (2019) 164 Supersymmetric solitons (2019) 207 , J. Phys. A , vol. 32, pp. 25–54, Springer (2020) [ , 07 Phys. Rev. B 109 , arXiv:1902.07227 , (2020) 104417 JHEP J. Phys. Condens. Matter Phys. Rev. D , , New localized solutions of the nonlinear field equations Front. in Phys. 101 , Introduction to Bessel Functions Solvable Models for Magnetic Skyrmions , JETP Lett. Nuclear Matrix Elements for ]. , . in QRPA SPIRE A IN [ Skyrmion line Publications (1958). easy-plane anisotropy Crystals Systems and Complexity micromagnetic framework Two-dimensional skyrmion bags in655 liquid crystals and ferromagnets metallic chiral magnets: Dynamics, pinning and creep Magnetic Film Quantum Theory and Symmetries g 247. stuck on impurities Phys. Rev. B Moduli matrix approach Mathematical Physics, Cambridge University Press (2009). supersymmetry M. Kobayashi and M. Nitta, F. Bowman, V.E. Timofeev, A.O. Sorokin and D.N. Aristov, A. Saxena, P.G. Kevrekidis and J. Cuevas-Maraver, D. Foster, C. Kind, P.J. Ackerman, J.-S.B. Tai, M.R. Dennis and I.I. Smalyukh, A.O. Leonov and I. Kézsmárki, S.-Z. Lin, C. Reichhardt, C.D. Batista and A. Saxena, D. Capic, D.A. Garanin and E.M. Chudnovsky, R. Brearton, G. van der Laan and T. Hesjedal, H. Ejiri, A. Bogdanov, C. Adam, K. Oles, T. Romanczukiewicz and A. Wereszczynski, M. Hongo, T. Fujimori, T. Misumi, M. Nitta and N.B. Sakai, Schroers, M. Shifman and A. Yung, C. Adam, J.M. Queiruga and A. Wereszczynski, M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, [40] [41] [38] [39] [36] [37] [33] [34] [35] [31] [32] [28] [29] [30] [26] [27] [25]