Neutral skyrmion configurations in the low-energy effective theory of spinor

condensate ferromagnets

R. W. Cherng1 and E. Demler1

1Physics Department, Harvard University, Cambridge, MA 02138

(Dated: December 11, 2009)

We study the low-energy effective theory of spinor condensate ferromagnets for the superfluidvelocity and magnetization degrees of freedom. This effective theory describes the competitionbetween spin stiffness and a long-ranged interaction between skyrmions, topological objects familiarfrom the theory of ordinary ferromagnets. We find exact solutions to the non-linear equationsof motion describing neutral configurations of skyrmions and anti-skyrmions. These analyticalsolutions provide a simple physical picture for the origin of crystalline magnetic order in spinorcondensate ferromagnets with dipolar interactions. We also point out the connections to effectivetheories for quantum Hall ferromagnets.

I. INTRODUCTION

For systems with broken symmetries, low-energy ef-fective theories provide a simple and powerful tool fordescribing the relevant degrees of freedom. The focus ison the long-wavelength Goldstone modes which emergefrom the microscopic degrees of freedom. This often re-veals the connections between seemingly unrelated sys-tems that share the same pattern of symmetry break-ing. The most well-known example is that of the com-plex scalar field used in the low-energy theories of bosonicsuperfluids, fermionic superconductors, and XY spin sys-tems.

Effective theories also bring to light subtle topologicaleffects that may become important at low energies. Gold-stone modes often carry a non-trivial topology which cangive rise to the appearance of topological defects. For ex-ample, in two spatial dimensions Kosterlitz and Thouless

FIG. 1: A single localized skyrmion carrying 4π net skyrmioncharge in the ordinary ferromagnet (left). Neutral configura-tion carrying zero net skyrmion charge in the spinor conden-sate ferromagnet (right). Notice the +z (−z) meron carrying+2π (+2π) net skyrmion charge at the origin (at infinity) forthe ordinary ferromagnet are mapped to a skyrmion (anti-skyrmions) carrying +4π (-4π) net skyrmion charge in thespinor condensate ferromagnet. Red (blue) background indi-cates positive (negative) skyrmion density q, black 2D arrowsthe superfluid velocity v, and shaded 3D arrows the magne-tization n.

pointed out the crucial role of vortices for the complexscalar field. Vortices are point-like topological defectsaround which the phase of complex scalar field winds byan integer multiple of 2π.

Ultracold atomic systems provide an ideal testingground for low-energy effective theories. The microscopicdegrees of freedom are well-isolated from the environmentexperimentally and well-understood theoretically. Thechallenge is in describing how these microscopic degreesof freedom organize at low energies in the presence ofnon-trivial interactions. When only one internal hyper-fine level is important, the phenomenon of scalar Bose-Einstein condensation is given by the low-energy theoryof the complex scalar field. A series of ground breakingexperiments have observed explicitly the role of vorticesfor two-dimensional condensates.

For ultracold atoms with a complex internal level struc-ture, there are various patterns of symmetry breaking.This lead to rich possibilities and challenges for low-energy effective descriptions. Spinor condensate ferro-magnets are one such system which has seen a numberof important experimental advancements. This includesthe development of optical dipole traps used for prepa-ration and phase-contrast imaging used for detection inS = 1 87Rb. In addition to the phase degree of freedomfamiliar from single-component condensates, the magne-tization naturally arises as a description of the low-energyspin degrees of freedom. A vector quantity sensitive toboth the population and coherences between the threehyperfine levels, the magnetization can be directly im-aged in experiments.

One of the most striking observations in spinor con-densate ferromagnets is the spontaneous formation ofcrystalline magnetic order. From an initial quasi-two-dimensional condensate prepared with a uniform mag-netization, a crystalline lattice of spin domains emergesspontaneously at sufficiently long times. The pres-ence of a condensate with magnetization spontaneouslybreaks global gauge invariance and spin rotational in-variance. Additionally, crystalline order for the mag-netization breaks real space translational and rotationalsymmetry. Previous works have pointed out the crucial

2

role of dipolar interactions in driving dynamical instabil-ities within the uniform condensate towards states withcrystalline order. Numerical analysis of the full multi-component Gross-Pitaevskii equations suggest dipolar in-teractions can give rise to states with crystalline order.

In this and a companion paper, we take a complemen-tary approach and focus on the low-energy effective the-ory of two-dimensional spinor condensate ferromagnets.This effective theory describes the interaction betweenthe superfluid velocity and the magnetization degrees offreedom. Previous work has derived the equations of mo-tion for this effective theory. We extend this result todemonstrate how the Lagrangian for the effective theorycan be written as a non-linear sigma model in terms ofthe magnetization alone. The effect of the superfluid ve-locity is to induce a long-ranged interaction term betweenskyrmions, topological defects familiar from the theory ofordinary ferromagnets. In contrast to the point-like topo-logical defects of vortices, skyrmions describe extendedmagnetization textures which carry a quantized topolog-ical charge.

In the companion paper, we study how symmetrygroups containing combined real space translational, realspace rotational, and spin space rotational symmetry op-erations can be used to classify possible crystalline mag-netic orders. Within each symmetry class, we find min-imal energy configurations describing non-trivial crys-talline configurations.

The main purpose of this paper is to give a simplephysical physical picture for the origin of crystalline or-der in terms of neutral configurations of skyrmions andanti-skyrmions. Long-ranged skyrmion interactions forcemagnetization configurations to have net neutral collec-tions of topological defects. We show this explicitlyby finding exact analytical solutions for the non-linearequations of motion describing both localized collectionsof skyrmions and anti-skyrmions as well as extendedskyrmion and anti-skyrmion stripes.

Skyrmions have non-trivial spin configurations thatspontaneously break translational and rotational invari-ance in real space. A neutral collection of such topolog-ical objects is able to take advantage of the dipolar in-teraction energy without a large penalty in the skyrmioninteraction energy. The exact solutions we find withoutdipolar interactions closely resemble the minimal energyconfigurations found numerically in the presence of dipo-lar interactions. The role of dipolar interactions can thenbe seen as stabilizing these non-trivial solutions of the ef-fective theory.

In addition, we point out the effective theory of spinorcondensate ferromagnets is essentially identical to thatof the quantum Hall ferromagnets. Although the mi-croscopic degrees of freedom are fermionic electrons, themagnetization order parameter is bosonic. The Coloumbinteraction between electrons then gives a contributionto the skyrmion interaction for the effective theory. Theresulting skyrmion interaction is qualitatively the sameas the one for spinor condensate ferromagnets. This sug-

gests the study of spinor condensate ferromagnets mayshed light on aspects of quantum hall ferromagnets andvice versa.

The plan of this paper is as follows. In Sec. II wereview the theory of ordinary ferromagnets and howskyrmion solutions arise from the non-linear Landau-Lifshitz equations of motion. We then proceed to re-view how these skyrmion solutions are used in the low-energy effective theory of quantum Hall ferromagnets inSec. III. Next we derive the low-energy effective the-ory for spinor condensate ferromagnets in Sec. IV. Inparticular, we demonstrate how a long-ranged skyrmioninteraction term (which also appears for quantum hallferromagnets) arises from coupling of the magnetizationto the superfluid velocity.

In Sec. V, we discuss how to interpret the mathemati-cal structure of skyrmion solutions for ordinary ferromag-nets in terms of a separation of variables. This approachallows us to use find new exact solutions for the oridi-nary ferromagnet. More importantly, it also allows us togeneralize the skyrmion solutions of the ordinary ferro-magnet to find exact solutions for the spinor condensateferromagnet. These latter solutions describe both neutralcollections of localized skyrmions and anti-skyrmions aswell as extended stripe configurations. Finally, we dis-cuss how the exact solutions we find offer insight intoquantum hall ferromagnets and spinor condensate ferro-magnets with dipolar interactions in Sec. VI

II. SKYRMIONS IN FERROMAGNETS

We begin by reviewing the theory of ordinary two-dimensional ferromagnets described by the following La-grangian, Hamiltonian, and Landau-Lifshitz equations ofmotion

L = −S∫

dtd2xA(n) · ∂tn−∫

dtH

H =S

4

∫

d2x∇(n)2

∂tn =1

2n×∇2n (1)

where n is a three component real unit vector and A(n)is the unit monopole vector potential. The order param-eter n describes the magnetization and is a unit vectorliving on the sphere. Calculating the variation of the La-grangian to derive the Landau-Lifshitz equations can bedone by using δn = δw × n. This is consistent with theconstraint |n| = 1 since δn · n = 0 by construction. Thevariation of the term

∫

dtA(n) · ∂tn is clearest when us-ing its geometric interpretation as the area on the sphereswept by n.

In addition to the trivial uniform solution, there arenon-trivial soliton solutions to the Landau-Lifshitz equa-tions called skyrmions. By parameterizing

n =[

sin(α) sin(β) sin(α) sin(β) cos(α)]T

(2)

3

we see α controls nz, the z component of the magnetiza-tion while β controls the canonically conjugate variablegiving the orientation of nx, ny, the x, y components ofthe magnetization.

Skyrmion solutions are given by tan(α/2)eiβ =exp[f(x+ iy)] where f(z) is a holomorphic function. Thefunction f(z) ∼ n log(z − z0) can have logarithmic sin-gularities. Since β has to be 2π periodic, the residuesof these singularities must be integers. This impliesexp[f(x + iy)] can only have zeros or poles. The result-ing spin configuration has nz = +1 (nz = −1) at ze-ros (poles) of exp[f(z)] while nx, ny wind anti-clockwise(clockwise) along a path circling the zeros (poles) in aanti-clockwise direction.

These solutions describe topological defects of ordinaryferromagnets. For fixed boundary conditions, smooth,finite energy configurations are separated into distinctclasses characterized by a quantized topological invariant∫

d2xq(x) = 4πN where N is the number of times ncovers the sphere. Here

q = ǫµν n · ∇µn×∇ν n (3)

is the skyrmion density which is positive for f(z) holo-morphic. From here on, lower Greek indices (upper Ro-man indices) refer to real space (order parameter) com-ponents.

Consider the single skyrmion solution shown in Fig.1. It corresponds to f(z) = log(z), carries net skyrmioncharge 4π, and can be decomposed into a +z meron atthe origin and a −z meron at infinity. Here a +z (−z)meron at z0 is a spin configuration carrying net skymrioncharge 2π that has nz = +1 (nz = −1) at z0 with nx, ny

winding anti-clockwise (clockwise) near z0.

Notice the Lagrangian in Eq. 1 has translational, ro-tational, and scale invariance in real space as well as ro-tational invariance in spin space. The single skyrmionsolution spontaneously breaks all of these symmetries.However, the action is invariant for f(z) = log[f0(z−z0)]with complex constants z0 and f0. Here z0 describes sin-gle skyrmion solutions related by real space translations,Re[f0] describes solutions related by scale transforma-tions, and Im[f0] describes solutions related by real spacerotations. In addition, the action is the same for Onwhere O is a rotation matrix describing solutions relatedby spin space transformations.

Collections of multiple skyrmions are described byf(z) having multiple singularities. A square lattice ofskyrmions with net skyrmion charge 8π per unit cell isshown in the left of Fig. 2. It can also be decomposedinto a collection of two +z merons and two −z meronsarranged antiferromagnetically. In addition, there aresolutions tan(α/2)eiβ = exp[f(x − iy)] with f(z) anti-holomorphic characterized by x, y components windingin the opposite direction as holomorphic solutions and qnegative.

FIG. 2: Unit cell for a lattice of localized skyrmions carrying8π net skyrmion charge per unit cell in the ordinary ferromag-net (left). Unit cell for a neutral configuration carrying zeronet skyrmion charge per unit cell in the spinor condensateferromagnet (right). Notice +z (−z) merons are mapped toskyrmions (anti-skyrmions). Red (blue) background indicatespositive (negative) skyrmion density q, black 2D arrows thesuperfluid velocity v, and shaded 3D arrows the magnetiza-tion n.

III. QUANTUM HALL FERROMAGNETS

After describing the structure of skyrmion solutions forordinary ferromagnets, we now briefly review how theyare used in the study of quantum Hall ferromagnets. Forthe low-energy effective theory of quantum Hall ferro-magnets, spin 1/2 electrons in a magnetic field are de-scribed by a two component Chern-Simons composite bo-son theory. The bosons couple to the physical gauge fielddue to te magnetic field and a fictitious Chern-Simonsgauge field which attaches appropriate flux quanta tochange from bosonic to fermionic statistics. At appro-priate filling fractions, the physical and Chern-Simonsfluxes cancel and the resulting quantum Hall plateau isdescribed as a condensate of composite bosons.

The Lagrangian and Hamiltonian for the resulting ef-fective theory is given by

L = − 1

2

∫

dtd2xA(n) · ∂tn−∫

dtH

H =

∫

d2x

[

1

8(∇n)2 +

1

2~B · n

]

+1

8

∫

d2xd2y[q(x) − q]G(x− y)[q(y) − q] (4)

where ~B is the magnetic field giving rise to a linear Zee-man shift, q(x) is the skyrmion density and q =

∫

d2xq(x)is the net skyrmion charge.

The coupling of the bosons to the Chern-Simons fieldgives rise to the skyrmion interaction term. At long dis-tances, the Chern-Simons field ties the skyrmion densityto the deviation of the physical electron density from itsbackground value. Thus, the skyrmion density inheritsthe Coloumb interaction for the electron density. Awayfrom quantum Hall plateaus, the background value q isnon-zero and the singular |x− y|−1 behavior in G(x− y)

4

forces finite energy configurations to have non-zero netskyrmion charge

Thus the focus is on static configurations carrying netskyrmion charge. The skyrmion solutions of the ordinaryferromagnet have non-zero q and provide a good qualita-tive description of quantum Hall ferromagnets away fromquantum Hall plateaus. For example, a sqaure lattice ofskyrmions as shown in Fig. 2 carries a net charge of8π per unit cell and can be used as a starting point forstudying configurations carrying finite q. Although suchsolutions take into account the spin stiffness and the long-ranged divergence of the skyrmion interaction, quantita-tive results require additional analysis arising from thedetailed form of skyrmion interaction and the linear Zee-man shift. This usually involves numerical minimizationof the Hamiltonian in Eq. 4.

IV. SPINOR CONDENSATE FERROMAGNETS

Having reviewed known results on ordinary and quan-tum Hall ferromagnets, we now consider spin S spinorcondensate ferromagnets described by the microscopiccondensate wavefunction Ψ, a 2S + 1 complex vector.The microscopic Gross-Pitaevskii Lagrangian is given by

L =

∫

dtiΨ†∂tΨ −∫

dtH−∫

dtHS

H =

∫

d2x

[

1

2m|∇Ψ|2 + g0(Ψ

†Ψ)2 + gs(Ψ

† ~FΨ)2]

(5)

where ~F are spin matrices and g0 > 0 gives the spin-independent contact interaction strength while gs < 0gives the spin-dependent contact interaction strengthwhich favors finite magnetization. Here, HS denotesadditional spin dependent interactions such as thequadratic Zeeman shift and dipolar interactions.

As was the case for quantum Hall ferromagnets, weexpect a simpler description to emerge at low energies.The resulting low-energy effective theory should only in-volve the condensate phase φ and local magnetizationn which describe the order parameters of the system.Previous work has shown this can be done at the levelof the equations of motion. In this section, we extendthis result to derive the Lagrangian and Hamiltonianfor the effective theory solely in terms of the magneti-zation. However, the skyrmion density acts as a sourceof voriticity for the superfluid velocity. Thus, the ef-fect of the superfluid phase is to induce a logarithmicvortex-vortex interaction between skyrmions. The re-sulting non-linear sigma model is essentially identical tothat of the quantum Hall ferromagnet but with skyrmioninteraction G(x− y) ∼ log(x− y) having logarithmic be-havior instead of |x− y|−1 behavior.

We begin by considering energies below the scale ofspin-independent g0 and ferromagnetic spin-dependent gs

contact interactions. The condensate has fixed density

Ψ†Ψ = ρ and fully polarized magnetization Ψ

† ~FΨ =

Sρn where ~F are spin matrices. The states that satisfythese constraints are parameterized solely in terms of thelow-energy degrees of freedom

Ψ =√ρeiφψn, n · ~Fψn = Sρψn (6)

where φ descrribes the phase of the condensate and ψn isa fully polarized unit spinor with n describing the orien-tation of the magnetization. Although φ is not directlyobservable, the superfluid velocity is a physical quantity

vµ = ∇µφ− iψ†n∇µψn (7)

which has contributions from both φ and ψn.In this paper, we are primarily interested in the com-

petition between the spin stiffness and superfluid ki-netic energy. In the companion paper, we address theeffect of dipolar interactions. From here on, we con-sider the case HS = 0. From the Gross-Pitaevskii La-grangian in Eq. 5, the Berry’s phase term becomesiΨ†∂tΨ = −SA(n) · ∂tn while the kinetic energy term isgiven by |∇Ψ|2 = S

2 (∇n)2+v2 and the interaction terms

give constants. This gives the Lagrangian and Hamilto-nian as

L = −S∫

dtd2xA(n) · ∂tn−∫

dtH

H =

∫

d2x

[

S

4(∇n)2 +

1

2v

2

]

(8)

where we take ρ = m = 1 for simplicity. Compared tothe Lagrangian describing ordinary ferromagnets in Eq.1, there is an additional superfluid kinetic energy termvµvµ.

The global phase φ enters the Lagrangian quadrati-cally and only through v. The equation of motion forφ gives ∇µvµ = 0 implying the superfluid velocity is di-vergenceless. This follows from v describing transport ofthe density ρ, a conserved quantity which is locally fixedat low energies due to the spin-independent contact in-teraction. This implies that vµvµ only depends on thedivergenceless part of v. In momentum space, this isvµ(+k)[δµν − kµkµ/k

2]vν(−k) which can also be writtenas Fµν(+k)Fµν(−k)/2k2.

Here we have introduce the analog of the field strengthtensor Fµν = ∇µvν−∇νvµ, a local quantity that dependsonly on the divergenceless part of v. In two dimensions,there is only one non-zero component to Fµν . From Eq.7, this is given by the skyrmion density FxY = −Fyx = q.Here we assume that the condensate phase φ has novortex-like singularities. This is valid at low energiessince otherwise the density ρ would have to vanish atthe vortex cores. These vortex cores are suppressed sincethey have an energetic cost which scales with log g0.

The above results give v solely in terms of n as

∇µvµ = 0, ǫµν∇µvν = Sq (9)

5

with q the skyrmion density. Gradients in the order pa-rameter n arise in part from phase gradients in the con-densate wavefunction Ψ. Topologically non-trivial mag-netization configurations can thus give rise to vorticitydescribed by a non-zero curl ǫµν∇µvν 6= 0.

By introducing the two-dimensional logarithmicGreen’s function −∇2G(x) = δ(x) we can write thesuperfluid kinetic energy Fµν(+k)Fµν(−k)/2k2 in realspace and obtain

L = −S∫

dtd2xA(n) · ∂tn−∫

dtH

H =S

4

∫

d2x(∇n)2 +S2

2

∫

d2xd2yq(x)G(x− y)q(y)

(10)

with the corresponding equations of motion given by

(∂t + vµ∇µ) n =1

2n×∇2n (11)

along with Eq. 9 for the superfluid velocity solved by

vµ = Sǫµν∇νΦ, −∇2Φ = q (12)

where Φ(x) =∫

d2xG(x−y)q(y) has the interpretation ofthe two-dimensional Coloumb potential associated withq.

Compared to the Landau-Lifshitz equations describingordinary ferromagnets in Eq. 1, the replacement ∂t →∂t + vµ∇µ describes the advection of the magnetizationby the superfluid velocity. This advective term arise fromvariation of the superfluid kinetic energy term in Eq. 8 orequivalently from the skyrmion interaction term in Eq.10.

Recall the skyrmion density q gives the vorticity forthe superfluid velocity v. Thus, the second term in theHamiltonian above gives the pairwise logarithmic inter-action energy between vortices. In the thermodynamiclimit, the logarithmic divergence of G(x−y) at large dis-tances forces finite energy configurations to have zero netskyrmion density

∫

d2xq(x) = 0.Notice Eq. 10 for spinor condensate and Eq. 4 for

quantum hall ferromagnets have the same form. Al-though G(x−y) behaves as log |x−y| for the former and|x−y|−1 for the latter, both give singular contributions atlong wavelengths. However, the important absence of afinite backgroud value q in the skyrmion interaction im-plies configurations for spinor condensate ferromagnetsmust have zero net skyrmion charge.

V. EXACT SOLUTIONS WITH NEUTRAL

SKYRMION CHARGE

Recall the analytical skyrmion solutions carrying net

charge for the ordinary ferromagnet offered insight intomore complicated case of quantum Hall ferromagnets

FIG. 3: For a holomorphic function f(x + iy) = u + iv, exactsolutions for both the ordinary and spinor condensate ferro-magnet have the magnetization n controlled by contour linesof the real u and imaginary v parts of f . Magnetization forthe single skyrmion (left) and neutral configuration (right) ofFig. 1. Hue indicates orientation of nx, ny components of themagnetization and brightness gives the nz component withwhite (black) indicating nz = +1 (nz = −1). Both solutionsarise from the holomorphic function f(x + iy)) = log(x + iy)with azimuthal (radial) contour lines shown in black for thereal part u (imaginary part v). Notice the contour lines aremutually orthogonal and nz (nx + iny) is constant on contourlines of u (v). In addition, the winding number of nx + iny

does not change for the single skyrmion on the left whereasfor the neutral configuration it changes sign upon crossing thenz = −1 contour on the right.

away from the quantum Hall plateau. For spinor con-densate ferromangets, we will show how net neutral solu-tions with skyrmions and anti-skyrmions without dipolarinteractions offer insight into the more complicated casewith dipolar interactions.

In this section, we find exact analytical solutionsfor spinor condensate ferromagnets with logarithmicskyrmion interactions in the absence of dipolar interac-tions. We study the effect of including dipolar interac-tions numerically after a symmetry analysis in a com-panion paper. The exact solutions we find here greatlyresemble the numerical solutions in the companion paper.As we discuss in Sec. VI, the interpretation of the ex-act solutions in terms of neutral collections of skyrmionsand anti-skyrmions offers physical insight into the morecomplicated numerical solutions of the companion paper.

To find exact solutions with zero net skyrmion charge,it is vital to include to effect of the skyrmion interac-tion term. Recall it is the long wavelength divergenceof this term that forces configurations to have zero netskyrmion charge. Although this cannot be done exactlyfor |x − y|−1 interactions as in quantum hall ferromag-nets, it is possible for logarithmic interactions as in spinorcondensate ferromagnets. Physically, this is because thelogarithmic interaction arises solely from the superfluidkinetic energy which is scale invariant just like the spinstiffness term.

We begin with the parameterization of n in Eq. 2 usedin the skyrmion solutions of the ordinary ferromagnet.Notice α, β provide a set of orthogonal coordinates for

6

the sphere describing the order parameter space of n. Forf(x + iy) = u(x, y) + iv(x, y) holomorphic, u(x, y) andv(x, y) provide a set of orthogonal coordinates for theplane describing real space. Skyrmion solutions are givenby α = 2 tan−1(eu) and β = v, which can be understoodas a separation of variables.

Notice α(u) is a function of u only while β(v) is afunction of v only. Each orthogonal coordinate of theorder parameter space α, β is a function of only one or-thogonal coordinate of real space u, v, respectively. Thereason why using u and v as coordinates is tractableis because they satisfy the Cauchy-Riemann equations∂xu = +∂yv, ∂yu = −∂xv. In particular, this implies∇u · ∇v = 0 meaning countour lines of constant u areperpendicular to countour lines of constant v as requiredfor orthogonal coordinates. In addition, both ∇2u = 0and ∇2v = 0 satisfy Laplace’s equation. The above twoidentities simplify expressions involving ∇2 which arisein the equations of motion. In particular, when changingvariables from (x, y) to (u, v), the Laplacian retains itsform ∂2

x + ∂2x ∝ ∂2

u + ∂2v .

An alternative interpretation of the above separationof variables is as follows. Given an arbitrary configura-tion for n, consider the contour lines of constant nz, the zcomponent of the magnetization. For contour lines withnz 6= ±1 that form closed curves, consider the windingnumber of nx + iny = sin(α)eiβ . For smooth configura-tions, this is a quantized integer that cannot change be-tween neighboring contours which do not cross nz 6= ±1.This implies the winding number is constant in regionsbetween contours with nz 6= ±1. Label different contoursof nz by u and the label coordinate along each contourby v.

In order to have a non-zero winding number, β musthave some dependence on v and the minimal one is β ∝ v.In principal, β can also depend on u and have some non-monotonic dependence on v, but linear dependence is insome sense the smoothest one compatible with non-zerowinding number. We see that skyrmion solutions canbe interpreted in the above manner along with the addi-tional condition that u and v are mutually othogonal andsatisfy Laplace’s equation. Physically, these additionalconditions on u and v can be understood as a conse-quence of minimizing the spin stiffness. The relationshipbetween contour lines, winding number, and the exactsolutions we discuss in this section is shown in Fig. 3.

With this viewpoint, we can now generalize theskyrmion solutions for ordinary ferromagnets and alsofind new solutions for spinor condensate ferromagnets.For f(x+ iy) = u(x, y) + iv(x, y) holomorphic, we take

α = α(u), β = kv (13)

for the parameterization of Eq. 2. Notice we include aconstant of proportionality β = kv instead of β = v. Re-call for ordinary ferromagnets, β = v and f(x+ iy) holo-morphic give skyrmion solutions with positive skyrmiondensity q while β = v and f(x− iy) antiholomorphic giveanti-skyrmion solutions with q. We can treat both types

of solutions with just f(x+ iy) holomorphic by allowingbeta = kv with k positive or negative.

There are two cases to consider for f(z). The first iswhen f(z) is a polynomial in z with no singularities. Thiswill turn out to describe skyrmion and anti-skyrmionstripe and domain wall configurations for both ordinaryand spinor condensate ferromagnets. The second case iswhen f(z) has singularities. Since n should be single-valued, β and thus kv can only have constant 2πN dis-continuities with N integer. This implies f(z) can onlyhave logarithmic singularities. These solutions will turnout to simply be the localized skyrmion configurations forordinary ferromagnets and neutral collections of localizedskyrmions and anti-skyrmions for spinor condensate fer-romagnets.

Next we consider Eq. 12 for the superfluid velocity v.We find q only depends on u and solve −∇2Φ = q for Φdepending only on u to arrive at

q = −k cos(α)′|∂zf |2, vz = iSk[C + cos(α)]∂zf (14)

where vz = vx − ivy, ∂zf = ∂xf − i∂yf and C is aconstant of integration physically describing a u inde-pendent constant contribution to the superfluid velocity.From here on primes denote derivatives with respect tou.

For ordinary and spinor condensate ferromagnets, theequations of motion in Eq. 1 and Eq. 11 reduce to

2α′′ = +k2 sin(2α) (15)

2α′′ = −4SCk2 sin(α) − (2S − 1)k2 sin(2α) (16)

respectively. Notice the equations of motion for ordinaryferromagnets are formally given by the S = 0 limit forspinor condensate ferromagnets. Recall the spin stiffnessterm scales linearly with S whereas the superfluid kineticenergy scales quadratically with S. For spinor condensateferromagnets in the limit S → 0, the superfluid kineticenergy is negligible compared to the spin stiffness andthe ordinary ferromagnet is recovered. From here on, weconsider the more general equation of motion for spinorcondensate ferromagnets.

Interpreting u as time, this equation is that of a clas-sical particle with coordinate α and momentum α′. Thetotal energy E = K + U is a constant of motion whereK = (α′)2/2 is the kinetic energy while the periodic po-tential U and equation of motion are

U(α) = −2SCk2 cos(α) − 2S − 1

4k2 cos(2α)

α′ =√

E − U(α) (17)

with S = 0 for ordinary ferromagnets. For this type ofsolution, the total energy calculated from the Hamilto-nian in Eq. 10 is given by

H = N

∫

dudvS

2

[

2S + 1 + 4C2S

4k2 − E + α′2

]

(18)

7

and similarly, S = 0 for the term in brackets for ordinaryferromagnets.

The integer N comes from changing variables (x, y)to (u, v) and taking into account each (u, v) may occurfor multiple (x, y). Mathematically, it is given by thedegree of f viewed as a map from the complex plane toitself C → C. For localized skyrmion solutions for theordinary ferromagnet, it physically corresponds to theskyrmion number. As an example, f(z) = log z for thesolutions shown in Fig. 1 and each value of (u, v) occursexactly once for (x, y) ranging over the plane and N = 1.For f(z) corresponding to the solutions shown in Fig. 1,each value of (u, v) occurs exactly twice for (x, y) rangingover the unit cell and N = 2.

Here we comment on the significance of the parametersC and E. Different values of C and E correspond to so-lutions with different boundary conditions. Since Eq. 16is a second order differential equation, we have to specifyboth α(u) and α′(u), with the latter given indirectly bythe constant of motion E. In addition C controls thebackground value of the superfluid velocity. These con-siderations mean that the total energy in Eq. 18 cannotbe directly compared for different C and E. In particu-lar, one should not consider minimizing the total energyH with respect to E and C.

A. Localized skyrmions and anti-skyrmions

We begin by considering the case of localizedskyrmions for ordinary ferromagnets and neutral collec-tions of localized skyrmions and anti-skyrmions for spinorcondensate ferromagnets. This corresponds to f(z) hav-ing logarithmic singularities. The singularities should beof integer magnitude and k should also be an integer.

Requiring a well-behaved, finite energy solution givesrise to several constraints. Consider Eq. 14 for theskyrmion density q and supefluid velocity v. Since∂zf diverges at the logarithmic singularities, we requirecos(α)′ and C + cos(α) to vanish. In addition, a finiteregion in (x, y) near logarithmic singularities is mappedto an infinite region in (u, v). The constant term in Eq.18 for the total energy is then integrated over an infiniteinterval. Thus, we also require it to vanish.

These constraints uniquely specify E, C and theasymptotic value α−∞. For spinor condensate ferromag-nets, E = k2(1 + 6S)/4, C = ±1, cos(α−∞) = ∓1 withS = 0 for ordinary ferromagnets. For C = ±1, we findfor spinor condensate ferromagnets the solution of Eq.16 and the total energy of Eq. 18 given by

α(u) = cos−1(∓1) − 2 cot−1(√

2S sinh(ku))

H = 4πNSk2

(

1 +2S tan−1(

√2S − 1)√

2S − 1

)

(19)

which describes either a 0 → 2π or −π → +π kink solu-

tion. For C = ±1 we find for ordinary ferromagnets

α(u) = cos−1(∓1) + 2 tan−1(eku)

H = 2πNSk2 (20)

which describes in contrast either a 0 → π or −π → 0kink solution.

Consider the classical mechanics problem describingthe evolution of α(u) in Eq. 17. For these solutions,E lies at the maximum giving rise to kink solutions.For ordinary ferromagnets, the kinks connect 0 → π or−π → 0 and carry net positive or negative skyrmioncharge, respectively. For spinor condensate ferromagnets,the kinks connect −π → +π or 0 → 2π and carry netneutral skyrmion charge. The neutral configurations con-sist of regions of oppositely charged skyrmion and anti-skyrmions. These regions are separated by lines wherethe skyrmion density q vanishes, the magnetization n isalong z, and the superfluid velocity v is large.

For f(z) having a finite number of logarithmic singu-larities, we can write

f(z) = log

[

f0

∏Na

n=1(z − an)∏Nb

m=1(z − bm)

]

(21)

with the degree N of the function f(z) given by N =max(Na, Nb). Here, an (bn) give the locations of +z(−z) merons each carrying net skyrmion charge 2π forthe ordinary ferromagnet. In contrast, an (bn) give thelocations of skyrmions (anti-skyrmions) each carrying netskyrmion charge +4π (−4π) for the spinor condensateferromagnet. We show the corresponding plots of n, q,and v in Fig. 1 for f(z) = log(z). The single skyrmionsolution for the ordinary ferromagnet with S = 0 is shownon the left and a neutral configuration of one skyrmionand one anti-skyrmion for the spinor condensate ferro-magnet with S = 1 is shown on the right.

For a periodic lattice of logarithmic singularities,

f(z) = log

[

f0

∏Na

n=1 ϑ(z − an, τ)∏Nb

m=1 ϑ(z − bm, τ)

]

(22)

where f0 is a constant, and 1, τ give the basis vectorsgenerating the lattice in complex form, and ϑ(z, τ) is theelliptic theta function. In the lattice case, Na = Nb and∑

an =∑

bn in order to have f(z) periodic. This re-striction comes from requiring f(z+n+mτ) = f(z) andusing the identity ϑ(z + n + mτ, τ) = exp(−πim2τ −2πimz)ϑ(z, τ). The degree N of the function f(z) perunit cell is given by N = Na = Nb. Again, an, bn givethe locations of merons (skyrmions or anti-skyrmions)for the ordinary (spinor condensate) ferromagnet. Fig.2 shows plots for f(z) having a lattice of logarithmicsingularities with N = Na = Nb = 2, f0 = 1, τ = i,a1 = −a2 = (1 + i)/2, b1 = −b2 = (1 − i)/2. Again theordinary (spinor condensate) ferromagnet is on the left(right).

8

FIG. 4: Neutral stripe configuration satisfying non-trivial cor-ner boundary conditions. Red (blue) background indicatespositive (negative) skyrmion density q, black 2D arrows thesuperfluid velocity v, and shaded 3D arrows the magnetiza-tion n.

B. Stripe configurations

Now we turn to case of stripe configurations describedby f(z) polynomial in z. The behavior of α(u) solutionscontrolled by the potential in Eq. 17 changes as E crossescritical points dU/dα = 0 of the potential. We focuson f(z) = iz with the corresponding solutions doublyperiodic and describing stripe configurations.

For completeness, we first briefly consider f(z) givenby higher order polynomials. The corresponding stripesolutions are no longer doubly periodic, but satisfy non-trivial boundary conditions. For example, we show themagnetization n, skyrmion density q, and superfluid ve-locity v for f(z) = iz2 in Fig. 4. This solution sat-isfies corner boundary conditions with zero normal com-ponent to both the superfluid velocity v and spin currentJ

iµ = ni

vµ − ǫijknj∇µnk/2.

The different types of behavior for α(u) are illustratedschematically along with the resulting configurations inFig. 5. E above the global maximum corresponds toα(u) monotonic in u which from here on we denote as M .This solution describes a periodic stripe solution with nz

varying over the entire range ±1. E at a local maximumcorresponds to a kink solution for α(u) connecting α1

to α2 denoted as Kα2

α1. This solution describes a single

domain wall configuration in nz and is the analog of thelocalized solution described earlier. E below a local max-imum corresponds to α(u) oscillating near a fixed valueα0 denoted as Oα0

. Finally, E below the global minimumis forbidden, denoted as F .

Notice that for S = 0, S = 1/2, S > 1/2, the cos(2α)term in the potential U(α) of Eq. 17 is negative, zero,and positive. For the montonic solutionsM , this does notaffect the qualitative behavior of the resulting periodicstripe configurations. For kink solutions K connecting

0 → 2π or −π → +π (0 → π or −π → 0), the result-ing single domain wall carrys zero net skyrmion charge(positive or negative skyrmion charge) for spinor con-densate (ordinary) ferromagnets. Oscillatory solutionsO also have different behavior with oscillations centeredabout nz ≈ ±1 (nz ≈ 0) for spinor condensate (ordinary)ferromagnets.

For ordinary ferromagnet with S = 0, we parametrizeE = k2(1+2δ)/4 and find the solution of Eq. 16 and thetotal energy of Eq. 18 given by

α(u) = π/2 ± am(

ku√

1 + δ, (1 + δ)−1)

H =Sk2

2

[

−δ2

+ θ(1 + δ)

]

(23)

where H is the total energy density given by the aver-aging H over the unit cell. Also, am(x,m) is the Jacobiamplitude function and we define

θ(m) = mRe[E(m−1)]

Re[K(m−1)](24)

with K (E) the complete elliptic integral of the first (sec-ond) kind. For δ < −1 the solution is forbidden F . For−1 ≤ δ < 0 there are two oscillatory solutions at the±π/2 minima O±π/2. For δ = 0 there are two kink solu-

tions K+π0 and K0

−π. For δ > 0 the solution is monotonicM . We show the classification of solutions along with thetotal energy density for ordinary ferromagnets in the bot-tom left of Fig. 6. Notice the solutions and total energydensity do not depend on C. This is because C entersthrough the superfluid velocity which is absent from theLagrangian for ordinary ferromagnets.

For spinor condensate ferromagnets with S = 1/2, weparameterize the constant of motion E = |C|k2(1 + 2δ)and find the solution of Eq. 16 and the total energydensity from Eq. 18 given by

α(u) = cos−1(C/|C|) + 2am(

ku√

|C|(1 + δ), (1 + δ)−1)

H = k2

[

(|C| − 1)2

8− δ|C|

2+ |C|θ(1 + δ)

]

(25)

depends only on |C|. For δ < −1 the solution is forbiddenF . For −1 ≤ δ < 0 there is one oscillatory solution Oπ

(O0) for C < 0 (C > 0). For δ = 0 there is one kinksolution K2π

0 (K+π−π ) for C < 0 (C > 0). For δ > 0 the

solution is monotonic M . We show the classification ofsolutions along with the total energy density for S = 1/2spinor condensate ferromagnets in the bottom right ofFig. 6.

For spinor condensate ferromagnets with S > 1/2 weparameterize the constant of motion E = (2S− 1)k2(1+2δ)/4 and the constant C = (2S − 1)γ/2S. With γ = tand δ = −1 + 2σt where σ = ±1 we find one solution forEq. 16 while the total energy density from Eq. 18 given

9

FIG. 5: Stripe configurations for different boundary conditions in S ≥ 1/2 spinor condensate (top row) and S = 0 ordinaryferromagnets (bottom row). Red (blue) background indicates positive (negative) skyrmion density q, black 2D arrows thesuperfluid velocity v, and shaded 3D arrows the magnetization n. Notice nx, ny (nz) wind horizontally (oscillate vertically).The first column shows schematic plots of the classical periodic potential U(α) controlling the evolution of the angular variableα for nz = cos(α). Labels for the corresonding type of solution are below the dashed lines indicating the corresponding constantof motion E. For E above the maximum of U , α(u) is montonic in the coordinate u giving rise to periodic stripe configurationslabeled M with nz covering the entire range ±1. For E at a potential maximum, kink solutions connecting the maxima α1,α2 give rise to single domain wall configurations in nz labeled Kα2

α1. For E below the maximum, α oscillates about a minimum

located at α0 giving rise to periodic stripe configuration with nz oscillating about cos(α0). Notice that monotonic solutions Mare qualitatively similar for both S ≥ 1/2 spinor condensate and S = 0 ordinary ferromagnets. In contrast, spinor condensate(ordinary) ferromagnets have a single (two distinct) 2π (π) kink solutions K centered about α = 0 (α = ±π/2). In addition,there is a single (two distinct) oscillatory solutions O also centered about α = 0 (α = ±π/2) for spinor condensate (ordinary)ferromagnets.

by

α(u) = Arg

[

− t− σ + 2√

t(t− σ)s(u) + ts(u)2

1 + σt− σts(u)2

]

H =Sk2

2[h0 + (2S − 1)j0] (26)

where we define the auxiliary function

s(u) = sin(ku√

(1 − σt)(2S − 1)) (27)

in the solution for α(u) and the functions

h0 =((2S − 1)t− σS)2

4S

j0 =

{

σt√t− σ 0 ≤ σt ≤ 1

0 otherwise(28)

for the total energy density. With γ = r sinh(t) andδ = −1 + 2r sinh(t) we find two solutions for Eq. 16 and

the total energy density from Eq. 18 given by

α±(u) =Arg

[

−cosh(t/4)e±iw(u) ∓ sinh(t/4)

sinh(t/4)e±iw(u) ∓ cosh(t/4)

]

H =Sk2

2[h+ (2S − 1)j ∓ (2S − 1)k] (29)

where we define the auxiliary function

w(u) =am

[

ku√

2r(2S − 1),1 + r − r cosh(t)

2r

]

(30)

10

-1 0 1 Γ

-1

0

1∆

S>1�2

OΠ O0

OΠ, O0

K02 Π K-Π

+Π

KΑ2 Π-Α,K-Α

+ΑM

F

-1 0 1 C

-1

0

1∆

S=1�2

OΠ O0

K02 Π K-Π

+ΠM

F-1 0 1 C

-1

0

1∆

S=0

O-Π�2,O+Π�2

K-Π0 ,K0

ΠM

F

FIG. 6: Classification of solutions (2D plots) and total energy density (3D plots) for stripe configurations given differentboundary conditions in S > 1/2 (top row) and S = 1/2 (bottom left) spinor condensate as well as S = 0 (bottom right)ordinary ferromagnets. Fig. 5 illustrates the corresponding configurations. γ controls a constant contribution to the superfluidvelocity. δ controls the energy of an associated classical mechanics problem giving the evolution of nz = cos(α), the z componentof the magnetization. Monotonic solutions M have nz covering the entire range ±1. Kink solutions Kα2

α1describe single domain

wall configurations with nz connecting cos(α1) to cos(α2). Oscillatory solutions Oα0have nz oscillating about cos(α0). Notice

kink solutions K always separate monotonic M from oscillatory O solutions. For S > 1/2 spinor condensate ferromagnets,notice the two distinct oscillatory and kink solutions for each γ, δ in the region near the origin which are absent for S = 1/2.For S = 0 ordinary ferromagnets, there is no dependence on C.

in the solutions for α(u) and the functions

h =1 − (2S − 1)δ

2+

(2S − 1)2β2

4S

j =Re[2

√rE(ω)]

Re[K(ω)/√r]

+

Re[2√r[cosh(t/2)2Π(− sinh(t/2)2, ω) −K(ω)]]

Re[K(ω)/√r]

k =

{

Re[πr sinh(t)]

Re[K(ω)√

2/r], Oπ, O0 phase

0 otherwise

ω =1 + r − r cosh(t)

2r(31)

for the total energy density where Π(m,n) is the completeelliptic integral of the third kind. We show the classifica-tion of solutions along with the total energy density forS > 1/2 spinor condensate ferromagnets in the top rowof Fig. 6. The boundaries between solutions of differenttypes are given by δ = −1 + 2β, δ = −1 − 2β, δ = β2.For increasing γ, notice kink solutions evolve from justone K2π

0 through a region with two K2π−αα and K+α

−α ,

to just one K+π−π for γ < −1, −1 < γ < +1, +1 < γ,

respectively. The kink solutions separate the monotonic

solutions M from the oscillatory solutions. For increas-ing γ, the oscillatory solutions also evolve from just oneOπ to a region with two Oπ and O0, to just one O0 forδ < −1 − 2γ, −1 + 2γ ≥ δ ≥ −1 − 2γ, δ < −1 + 2γ,respectively.

VI. DISCUSSION

Having presented a unified description of both local-ized and extended stripe solutions in ordinary and spinorcondensate ferromagnets, we now turn to how these solu-tions offer insight into different physical phenomena. Wefirst consider quantum Hall systems. From Sec. III as anumber of previous works have discussed, configurationsfor quantum Hall ferromagnets away from quantum Hallplateaus carry net skymrion charge. Thus, solutions forthe ordinary ferromagnet describing collections of local-ized skyrmions carrying net charge as shown in the left ofFigs. 1 and 2 have been used extensively in this regime.

However, we showed in Sec. V that these solutions oflocalized topological objects can be derived in a unifiedframework along with extended stripe solutions. Therehave been a number of papers on the possibility of quan-tum Hall states with stripe order. At high Landau levels

11

FIG. 7: Numerically optimized configuration for two-dimensional spinor condensate ferromagnets with an effectivedipolar interaction modified by rapid Larmor precession. Themagnetic field B = x inducing Larmor precession lies in theplane. Lattice constants are a = 41µm and b = 45µm. Red(blue) background indicates positive (negative) skyrmion den-sity q, black 2D arrows the superfluid velocity v, and shaded3D arrows the magnetization n.

and with frozen spin degrees of freedom, Coloumb in-teraction may directly favor charge density waves. Suchstates are not directly comparable to the stripe solutionswe describe which have fixed total density and stripe or-der in the relative density. However, stripe order hasalso been proposed in the context of quantum Hall bilay-ers. Here, even though the total density between layersis fixed, both interlayer coherence and relative densityimbalance can develop. The isospin degree of freedomthat arises can be used to define an appropriate mag-netization vector n. Here, the phase of the interlayercoherence gives the orientation of nx, ny, while the rel-ative density imbalance gives nz. States with skyrmionstripe order and winding nx, ny have been proposed thatare direct analogs of the configurations shown in the toprow of Fig. 5.

For spinor condensate ferromagnets, experiments atBerkeley suggest the possibility of a condensate withcrystalline magnetic order. This crystalline order arisesfrom an effective dipolar interactions modified by rapidLarmor precession and reduced dimensionality. It candrive dynamical instabilities of the uniform state whichoccur in a characteristic pattern. Modes controlling thecomponent n parallel (perpendicular) to the magneticfield in spin space are unstable along wavevectors per-pendicular (parallel) to the magnetic field in real space.Instabilities of this type can give rise to the spin texturesshown in the stripe solutions of the bottom row in Fig.

5. Here, nz is modulated along the y direction while nx,ny wind along the x direction.

In a companion paper, we have performed a system-atic numerical study of minimal energy configurationsfor spinor condensate ferromagnets with dipolar interac-tions. This is made possible by the use of symmetry oper-ations combining real space and spin space operations todistinguish different symmetry classes of solutions. Forapplied magnetic field in the plane B = x correspond-ing to current experiments, we show the lowest energyconfiguration in Fig. 7.

Notice nz is modulated between +1 . . . − 1 just as inthe monotonic M solutions shown in Fig. 5. In addition,the nx, ny components wind along the x axis. However,notice the winding in the nx, ny components is not uni-form as in the solutions we find in this paper. In addition,the winding changes from clockwise to counter-clockwisehalfway along the x axis. For the solutions we find inthis paper, the skyrmion density forms stripes of oppo-site charge parallel to the x axis. For the lowest energyconfiguration in Fig. 7, the non-uniform winding leadsto concentration of the skyrmion density in smaller re-gions and modulation in the sign of the skyrmion densityalong the x axis. We find minimal energy configurationsin other symmetry classes are generally of this type withnz between oscillating between ±1 + 1 along y and nx,ny winding along x. However, the detailed form of thewinding along x varies for different classes.

Thus we see that the exact solutions for spinor conden-sate ferromagnets without dipolar interactions providesa more transparent physical picture for the numerical so-lutions with dipolar interactions. This can be seen as fol-lows. The solutions we find in this paper describe overallneutral collections of skyrmion and anti-skyrmion topo-logical objects. The neutrality constraint comes from thelong-ranged divergence of the skyrmion interaction andremains even when considering additional spin interac-tions such as the dipolar interaction. Morever, skyrmionsand anti-skyrmions themselves have a non-trivial spintexture which is evident in Fig. 5 showing the solutions ofthis paper. When dipolar interactions are included, suchspin textures can take advantage of the gain in dipolar in-teraction energy without chaning their qualitative struc-ture. However, quantitative details for minimal energyconfigurations such as the one shown in Fig. 7 requiredetailed analysis of the competition between dipolar in-teractions, skyrmion interactions, and spin stiffness.

In conclusion, we have presented the low-energy effec-tive theory of spinor condensate ferromagnets. This ef-fective theory describes the superfluid velocity and mag-netization degrees of freedom and can be written as anon-linear sigma model with long-ranged interactions be-tween skyrmions, the topological objects of the theory.Quantum Hall ferromagnets share a similar effective the-ory with long-ranged skyrmion interactions. For the caseof spinor condensate ferromagnets, we find exact solu-tions for the non-linear equations of motion describingneutral configurations of skyrmions and anti- skyrmions

12

carrying zero net skyrmion charge. These solutions de-scribe within a unified framework both collections of lo-calized topological objects as well as extended stripe con-figurations. In particular, they can be used to understandaspects of non-trivial spin textures in both quantum Hallferromagnets as well as spinor condensate ferromagnets

with dipolar interactions.

Acknowledgments