Vortex Molecules in Spinor Condensates
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Vortex Molecules in Spinor Condensates
Ari M. Turner ∗† and Eugene Demler ∗ ∗ Department of Physics, Harvard University, Cambridge MA 02138 and † Department of Physics, University of California, Berkeley CA 94720 (Dated: November 21, 2018)Condensates of atoms with spins can have vortices of several types; these are related to the sym-metry group of the atoms’ ground state. We discuss how, when a condensate is placed in a smallmagnetic field that breaks the spin symmetry, these vortices may form bound states. Using sym-metry classification of vortex-charge and rough estimates for vortex interactions, one can show thatsome configurations that are stable at zero temperature can decay at finite temperatures by crossingover energy barriers. Our focus is cyclic spin 2 condensates, which have tetrahedral symmetry.
In an image of a nematic liquid crystal by polarizedlight, one can identify topological defects of various topo-logical charges (see Ref. [1]). Bose condensates (see thebooks[2, 3]) are starting to provide another context forstudying topological defects: in the texture formed bythe phase and spin of a spinor condensate[4, 5, 6]. (Seealso [7, 8] for reviews of the theory and experimentaltechniques being applied to spinor condensates.)A topological defect in the phase of a superfluid isa quantized vortex. The discontinuity in the phase asthe defect is encircled must be 2 πn for an integer n ,and the circulation of the vortex is then n hm . In asingle-component superfluid, multiply quantized vortices( | n | >
1) are usually not stable. The widely known ex-planation is that the energy of a vortex is proportional to n . Thus a doubly quantized vortex ( n = 4) can lowerits energy by splitting into two singly quantized vortices.Similar arguments can be formulated for multicompo-nent condensates, but we will find that some vortices inthese condensates can be very long-lived in spite of hav-ing large energies. Our predictions are about condensatesof atoms with spin in which the rotational symmetry hasbeen weakly disrupted by a small parameter q , such asthe interaction between the spins and the magnetic field.(We focus on the cyclic condensates of spin 2 atoms, seeRef. [9].)Long-lived multiply quantized vortices are particularexamples of composite vortices, which are made up ofseveral vortices bound together. Ground states withcomplicated symmetries have many types of vortices[10].Symmetry violating fields provide a force that can bindsome groups types vortices together so that they forma “composite core” for a larger vortex. The compos-ite vortex might have two quanta of circulation (or havesome other extra-large topological charge). Such a vortexwould be surprising if observed under a low resolution,because the smaller vortices that make it up would behidden in its core; the behavior of the order parameterat infinity is determined entirely by the net topologicalcharge.Some earlier theories about vortices in multicompo-nent condensates also describe vortices with asymmetri-cal cores which can be regarded as collections of closelyspaced vortices. The parameter q makes this interpreta-tion even more meaningful: when q is very small, we find that the component vortices move far apart, so that theircores do not overlap, while still remaining bound. Ref.[11] describes vortices that can occur in a condensate oftwo atomic states when there is an RF-field producingcoherent transitions between the states. The binding ofthese vortices also comes from an asymmetry, but theasymmetry comes from the dependence of the interac-tion strength on the internal states of the atoms, ratherthan from an external magnetic field. Ref. [12] stud-ies vortices of spinor atoms in a magnetic field, like us,but focuses on spin 1, and describes composite vorticesas well. Since the scenario involves rotation as well asa magnetic field, the vortices would be held close to the axis of the condensate by rotational confinement withoutthe magnetic field. With just a magnetic field, we showthat the vortices are attracted to one another .Vortices and bound states are not hard to picture, bytaking advantage of the fact that the state of a spinoratom can be represented by a geometrical figure. Theappropriate shape depends on the type of condensate.For a ferromagnetic condensate, a stake pointing in thedirection of the magnetization could represent the lo-cal state of the condensate. Other condensates can berepresented by more complicated shapes. Now imaginea plane filled with identical shapes (tetrahedra, for thecyclic phase), with orientations varying continuously as afunction of position. This shape field (or “spin texture”)together with a phase field would represent a nonuniformstate of a condensate. If the shapes rotate around a fixedsymmetry axis as some point is encircled then the spintexture has a topological defect at this point. Such con-figurations generalize vortices, because some of them areaccompanied by persistent spin or charge currents. Foreach symmetry of the tetrahedron, there will be a vortexwhen the Hamiltonian is SU symmetric. (Each discretesubgroup of SU describes the vortices of some phase foratoms of some spin[13]; to find vortices for more evenmore complicated groups like SO or SO , one mightwant to study gases of spin atoms[14, 15].) The vortexspectrum is decimated when a magnetic field is applied,since the field favors a particular orientation of the orderparameter –e.g., tetrahedra may want to have an order-three axis aligned with the magnetic field. The spec-trum of vortices is then reduced from the full spectrumof “tetrahedral” vortices (based on arbitrary symmetriesof the tetrahedron) to “field-aligned” vortices, where thetetrahedra must rotate around this order three axis, so asnot to lose their alignment with the magnetic field. Wemay introduce the ground state space M q (of tetrahedraaligned with the magnetic field), where the interactionenergy is V = V min , and the space of ground states of the SU invariant part of the Hamiltonian, M (of tetrahedrawith arbitrary orientations). A set of vortices can be as-signed a topological charge based on the loop traced outin one of these spaces by the values of the order param-eter on a circle around the set. Specifically, the topolog-ical charge describes the symmetry transformation thatbrings the order parameter back to its initial value as theset of vortices is circumambulated. (The relation betweentopology and symmetry is presented in [10].) From thegeneral point of view, the reduction of the charge typesfrom the tetrahedral to the field-aligned ones when q isturned on results because the smaller ground state spaceat nonzero q , M q , has fewer closed loops[52].If q is very small, the excess energy of an order pa-rameter in M rather than M q is small. There is thena hierarchy of spaces M q , M and H (the whole Hilbertspace, corresponding to arbitrarily distorted tetrahedra)with increasing energy scales. When q = 0, a vortexhas to have a field-aligned charge at infinity because thetetrahedra must eventually move into M q to avoid toobig of an energy cost. But if q is small, the hierarchy ofthe order parameter space is reflected in the fact that avortex with a field-aligned charge can have a composite core that may contain tetrahedral vortices, as in Fig. 1;the core is sort of like the pulp and seeds of a fruit. Insidethe fruit is a texture of arbitrarily oriented tetrahedra,almost as if q were equal to zero. The seeds are thenthe cores of tetrahedral vortices and the pulp is qualita-tively the same as the texture that would surround thesevortices in the absence of the magnetic field. The netcharge of the vortices has to be field-aligned so that theloop of order parameters that surrounds the whole corecan move into M q as r → ∞ . This picture becomesmore accurate as q becomes small, since the vortex coreswithin the composite core are far apart in that case– thesize of a region where the order parameter is in M − M q or H − M is inversely proportional to the energy scalefor each space. As q →
0, the binding of the tetrahedralcharges becomes weaker and weaker, until they becomefree from each other; each vortex therefore is describedby independent degrees of freedom when q = 0.The optimal size L q of a composite vortex results fromcompetition between two forces–a confinement force fromthe anisotropy term and the familiar logarithmic inter-action of vortices. The symmetry breaking energy fa-vors minimizing the area over which the order parameterleaves M q , pushing the component vortices toward oneanother. On the other hand, it cannot compress them toa point since the Coulomb-like repulsion one expects ofvortices keeps them apart.Some of these vortex molecules will turn out to bemetastable. Ref. [12] mentions an interesting clue to PSfrag replacements Γ Q FIG. 1: A composite vortex reflects the hierarchy of the or-der parameter space. In the white, grey, and black regionsthe order parameter moves from M q to M to H which hasthe highest energy of all. The charges of the subvortices aretetrahedral charges, represented by Γ and the charges of thecomposite vortex is a “field-aligned” charge. such a phenomenon; namely there are multiple steadystate wave functions describing a condensate with a givenmagnetization and rotational frequency; these local min-ima of the energy function can maybe be analyzed usingthe group theoretic metastability conditions we discussin Section III C. For a spinor condensate, wave functionsfor states besides the ground state are experimentallyimportant, since the experiments of Ref. [16], as well asthe liquid crystal experiments of Ref. [17], reveal compli-cated textures produced by chance; an initial fluctuationaround a uniform excited state becomes unstable andevolves into an intricate nonequilibrium texture. So it isuseful to analyze spin textures which are only local min-ima of the Gross-Pitaevskii energy functional (like themetastable vortex molecules considered here) as well asunstable equilibria (which take a long time to fluctuateout of their initial configuration). Examples of unstableequilibrium are described for single-component conden-sates by [18]. The process by which textures form outof uniform initial states has been discussed in theoreticalarticles, including [19, 20] (on the statistics of the spinfluctuations and vortices that are produced from this ran-dom evolution), [21] (on the spectrum of instabilities) and[22, 23] on the dynamics of spinor condensates. The ex-periments described in [24] show that the patterns thatevolve in rubidium condensates are probably affected bydipole-dipole interactions, though we are not consideringthese. Dipole-dipole interactions lead to antiferromag-netic phases[25], which maybe can be described as groundstate configurations of vortices.Besides just hoping for unusual types of vortices to PSfrag replacements a) b) xyz
FIG. 2: Geometrical representation of the tetrahedral super-fluid phases of spin 2 atoms. Two orientations of a tetra-hedron, corresponding to the spin roots of the ground statespinor, are illustrated and the symmetry axes are labelled.(The orientation of the coordinate axes, shown on the side, isthe same in both cases.) a) An orientation with the vertical z -axis along an order three axis, corresponding to the groundstate √ n χ (see Eq. (2)). b) An orientation with the z -axisalong an order two axis, corresponding to the ground state √ n χ (see Eq. (29)). For a magnetic field along the z axis,orientation a) is preferred when c > c <
4. Note that the three order 2 axes, la-belled
A, B, C can be used as a set of three orthogonal bodyaxes. form, one can make a vortex lattice by rotating a con-densate. The effects of rotating on a spinor condensatehave been investigated theoretically in [26] as well as thereview [27]. Experiments can also make a single vortex ofa prescribed type[28, 29, 30]. Excited by these possibil-ities, physicists have come up with several types of vor-tices and topological defects they would be interested inseeing: skyrmions[31, 32], monopoles[33], textures whoseorder parameter-field lines make linked loops[34], as wellas the noncommutative vortices of the cyclic phase thatwe will be expanding on here[35].The key to our discussion of vortices will bea geometrical representation of the order parame-ter, allowing us to visualize a texture of the cyclicphase as a field of tetrahedra with different orien-tations. (See Fig. 2.) Without this represen-tation, a spin texture would be given by a spinorfield ( ψ ( x, y ) , ψ ( x, y ) , ψ ( x, y ) , ψ − ( x, y ) , ψ − ( x, y )) T ;the fact that this spinor lies in the ground state manifold M would have to be described by a set of polynomialrelations between the five components. A more reveal-ing way to represent a spinor is to draw a geometricalfigure consisting of “spin-roots” (as in [36]), and in thecyclic phase, these spin roots form the vertices of a tetra-hedron. (A similar construction can be used to classifyvortices in condensates of spin 3 atoms, see Ref. [37].)Even without using the spin-root interpretation, one canjustify using tetrahedra to represent order parameters inthe cyclic phase because they are a concrete way of rep- resenting the symmetry of this phase. Ref. [35] workedout the symmetry group of a state in the cyclic phase byfinding all the pairs ˆ n , α such that e − iα F · ˆ n χ ∝ χ (1)where χ = q q . (2)The spinor √ n χ is a representative cyclic state, if n is the density of the condensate. These symmetries arethe same as the symmetries of a tetrahedron orientedas in Fig. 2a; hence we may represent the state √ n χ by this tetrahedron. Any other ground state should berepresented by a tetrahedron oriented so that its symme-try axes and symmetry axes of the spinor coincide. Theappropriate orientation of the tetrahedron for a givenground state can be determined in an automatic way bycalculating the spin roots.The Hamiltonian for spin 2 atoms in a magneticfield, H = RR d u [ ~ m ∇ ψ † ∇ ψ + V tot ( ψ )] has a simpleexpression[9] in terms of the density n = ψ † ψ , magneti-zation m = ψ † F ψ , and singlet-pair amplitude θ = ψ † t ψ. ( ψ t stands for the time reversal of ψ .) V tot ( ψ ) = 12 ( αn + β m + cβ | θ | ) − qψ † F z ψ − µψ † ψ (3)The first three terms describe the rotationally symmet-ric interactions of pairs of atoms. The first one de-scribes repulsion between a pair of atoms and the nexttwo terms describe additional, smaller interactions, thatdepend on the spin states of the two colliding atoms.These terms determine the spinor ground state in theabsence of a magnetic field[9]. (The spin-dependent in-teraction strengths β, cβ can be expressed in terms ofthe scattering lengths.) The properties of spin 2 atoms,which are described by this Hamiltonian, and of spin1 atoms have been investigated experimentally in Refs.[16, 38, 39, 40, 41]; [42] reviews more experimental phe-nomena. Ref. [39, 40] found values for α, β and c for Rb that are consistent with theoretical predictions, al-though even the sign of c is not known for sure, because c is small. The final term contains the chemical potential µ . If β and c are positive, the ground state of a conden-sate of spin 2 atoms is cyclic. The deformation of a cyclicstate due to a magnetic field is the simplest if we assumethat α ≫ | β | and that c is on the order of 1. We there-fore assume c ∼
1, though c ≪ B along the z -axisand q ∝ B . (See [2] for the explanation of why thequadratic Zeeman term q but not the linear term is rel-evant if the condensate’s initial magnetization is zero. Anonzero magnetization is described by a Lagrange multi-plier term − p RR d x ψ † F z ψ , which looks like a linear Zee-man coupling. We assume p = 0; a nonzero p should havesimilar consequences as a nonzero q , since both break therotational symmetry.)Introducing a small q reduces the ground state spacefrom M to M q . This can be explained (see Sec. II A)by finding the modulation of the energy of an arbitrarytetrahedral state as a function of the orientation of thecorresponding tetrahedron: V eff = ( c −
4) 3 q cβ (cos α + cos α + cos α ) , (4)where α , α and α are the angles between the z -axisand three body-axes A, B, C fixed to the tetrahedron (seeFig. 2a). The orientations that minimize this energy arethe true ground states at nonzero q . If c > z -axis perpendicular to a face as in Fig.2a, with a ground state energy of V min = ( c − q cβ . Thusthe absolute ground state space M q = M q containsall the wave functions that are arbitrary rotations aboutthe z -axis (combined with rephasings) of √ n χ . When c <
4, the ground states M q are rotations about z of thetetrahedron illustrated in Fig. 2b. In particular, when q = 0, there is a phase transition at c = 4, though thereis nothing special about c = 4 in zero magnetic field. Inthis paper, we will mostly assume that c > ◦ of the ground state tetrahedron (seeFig. 2a) about the A axis, ( q , , q ). Such a vortexis described at large distances by ψ ( r, φ ) = e − i φ √ ( √ F x + F z ) √ n χ (5)where r, φ are polar coordinates centered on the core ofthe vortex. This vortex has an excess energy (relativeto the ground state energy of the condensate) which di-verges with the condensate size. In fact, its Zeeman en-ergy density (with V min subtracted) is given according toEq. (4) by[53] ( c − q )6 βc (sin φ ) . (6)The tetrahedron is pointing in the wrong direction exceptalong the positive and negative x -axis; and the integratedenergy is proportional to the area: E misalign ∼ q R βc (7)where R is the condensate’s radius. Such a vortex cannotbe the only vortex in an infinite condensate! K.E.Z.E.
FIG. 3: Illustration of the energy costs in a vortex molecule.The energy in the region occupied by the two vortices is dom-inated by the quadratic Zeeman cost and the energy outsideit is dominated by kinetic energy. Increasing the molecule sizedecreases the kinetic energy and increases the Zeeman energy.The equilibrium size L is determined by minimizing the sum. The vortex described by Eq. (5) can form a partner-ship with another vortex of the same type, producinga molecule that can exist without costing too much en-ergy. This is because the net charge of the two vorticesis compatible with the magnetic field. The combinationof two 180 ◦ rotations of the tetrahedral order parameterabout the A axis is a 360 ◦ rotation about the A -axis, butsince any rotation axis is a 360 ◦ symmetry of the tetrahe-dron, the rotation axis on circles of radius r enclosing thevortices can be tilted continuously relative to the tetra-hedron as r increases until it becomes the R axis instead.Then the tetrahedra align with the magnetic field farfrom the vortices and stay in M q . Each vortex screensthe part of the other vortex’s charge that produces thelarge Zeeman cost, as illustrated in Fig. 3. The Zeemanenergy in the region around the vortices, where the tetra-hedra are still tilted, may be estimated by replacing thetotal condensate size R in Eq. (7) by the diameter L ofthis region. The Zeeman energy tries to pull the vorticestoward one another but the elastic energy cost of rapidchanges in the order parameter (the gradient term in theHamiltonian) opposes this tendency: The tetrahedra ro-tate twice as fast around circles beyond L , where the twovortices act in concert. Therefore the elastic energy in-creases as L becomes smaller; this leads to the Coulombrepulsion between the vortices, 2 π n ~ m ln Ra − π n ~ m ln La c ,where m is the mass of the atoms in the condensate. Theenergy of the vortex molecule is therefore E = k q L β − π n ~ m ln La c + cnst. (8)where k is a numerical constant. The equilibrium sizecan be determined by minimizing over L . The quadraticZeeman force binds this molecule together while the re-pulsion keeps the vortices from merging.If the vortices were to coalesce, then they could reactto form a set of vortices that are not bound by the Zee-man energy; one possible set of decay products is threevortices each involving a rotation through 120 ◦ aboutthe R axis (which have the same net 360 ◦ rotation as theoriginal pair of vortices). The vortex molecule of the two A rotations is metastable because thermal fluctuationsmay overcome their Coulomb repulsion and push themtogether, leading to such a fission process.The rest of this paper elaborates: it gives a qualita-tively correct expression for the spin texture surround-ing the molecule just described and determines how thismolecule actually decays. In order to do this, we willgive some more general results: first Section I, summa-rizes the non-commutative group theory of combiningvortex charges and a classification of the tetrahedral vor-tices (the vortices that occur by themselves when q = 0and in clusters when q = 0); then Section II estimatesthe elastic energies and Zeeman energies of such clustersas functions of these vortex charges; finally Section IIIgives criteria determining which types of tetrahedral vor-tices form bound states or metastable states (see SectionIII C). The last two sections illustrate the criteria with afew additional surprising examples (see Section IV) andgive some basic ideas about how to observe metastablevortices (Section V). I. TOPOLOGICAL CHARGES
Vortices are simplest to understand for an interactionenergy that has a single manifold, N , of ground statesand no hierarchy. The order parameter must move into N far away from any spin texture. Vortices are classifiedby the topology of the circuit traced out in N by theorder parameter on a large circle containing a vortex orset of vortices. Although there are many ways to addwiggles to a given circuit, only the topological structureis important, leading to a discrete set of possible vortexcharges. Fig. 4 shows how a circuit may be tangled withholes in the space N . As a vortex evolves, such a circuitcan evolve only into other circuits that are tangled in thesame way; thus the vortex charge is conserved.This generalizes circulation-conservation in a single-component condensate: The circulation quantum num-ber n for a set of vortices is also the number of times thatthe wave function at infinity winds around the circle thatminimizes the Mexican hat potential[43]. The tanglingin a multicomponent condensate can be described by agroup of generalized winding numbers around the groundstate manifold N , π ( N ), the “fundamental group” (seeFig. 4).Besides the conservation of topological charge, two fur-ther properties of vortices follow from the geometry of theinternal ground-state space. First of all, the net winding or topological “charge” of a set of vortices can be foundby multiplying their charges together, using the definitionof multiplication for the fundamental group. (In a gen-eral space, the fundamental group has a multiplicationoperation defined by splicing circuits together.) This ruleis the generalization of adding the n ’s of the individualvortices in the scalar order-parameter case. (The funda- PSfrag replacements N ΓFIG. 4: The order parameter at infinity of a vortex windsaround a loop Γ in the order parameter space M . Realisticorder parameter spaces are usually more symmetrical. mental group is noncommutative so one has to multiplythe charges together in the right order, see Appendix B.)Secondly, the energy of a vortex can be estimated as afunction of the winding behavior at infinity. For a vortexthat minimizes this energy, the order parameter travelsalong a geodesic in N . Since the interaction energy V is constant (and equal to its minimum V min ) at infinity,the energy of a vortex is determined by the elastic en-ergy, that is, the cost of variations in ψ as a functionof the azimuthal angle φ . The closed loop traced outby ψ ( R cos φ, R sin φ ) will relax to make this energy-costsmall. When it shrinks as much as is possible withoutleaving N , it becomes a geodesic in N . The energy ofthe vortex is related to the length l of this geodesic by E = ( ~ n m ln Ra c ) l π (9)where R and a c are the radii of the condensate and thevortex core. (Note that replacing l by 2 πn gives the stan-dard expression for a vortex in a scalar condensate.)A small magnetic field introduces hierarchies into theorder parameter space, leading to spin textures that windaround one manifold at an intermediate length scale andaround a smaller manifold far away. A. Spin Textures and their Vortices
Let us consider how nested vortices hold themselvestogether in a continuous “spin texture.” We will try togive a general argument showing how topology and theenergy-hierarchy of the order parameter spaces impliesthat any texture is made up of a set of composite vorticeswhich in turn are made up of clusters of nearly point-likevortices.The order parameter subspaces in order of increas-ing energies are the two-dimensional M q , the four-dimensional M and the 4 F + 2 dimensional H (for pi2pi3 pipi1A Bphi0 phi2pi PSfrag replacements φ = 0 φ = 2 π a) b) Γ Γ Γ Qλ FIG. 5: Classifying vortex topologies. a) The combined topol-ogy of a set of vortices can be found be seeing how the orderparameter changes around a closed loop parameterized by φ .The loop is drawn with a small gap between φ = 0 and φ = 2 π to indicate that g ( φ ) and θ ( φ ) vary continuously only between0 and 2 π . The topological charge is defined as the jump inthe values of g and θ across the gap. b) Vortices that cannotexist by themselves in the presence of a magnetic field can stillsometimes appear in clusters. The charges Γ , Γ ,Γ in one ofthe clusters can involve rotations around arbitrary axes, butthe combined charge Q has to use the direction of the mag-netic field for its rotational axis to avoid a large energy-cost. spin F ), with corresponding energy scales V − V min =0 , ǫ ( q ) = q β (see Eq. (4)) and βn (see Eq. (3); as longas the density of the condensate does not vary, β ratherthan α sets the energy scale). The condensate can moveout of the ground state into one of the higher-energysubspaces if forced to by topology, spending less spacein the manifolds with the greater energies. (This is likesaving eggnog–with all that fat!–for one week out of theyear.)For a particular subspace M i in this sequence, inwhich V ( ψ ) − V min ∼ ǫ i , the size L i of the regions where ψ is in M i is typically L i ∼ q Kǫ i , where K = ~ n m isthe elasticity of the condensate. This relation can beguessed at without understanding anything about thefield configuration in such a region; just assume equipar-tition between kinetic and interaction energies, so that ǫ i is equal to KL i , a typical scale for the kinetic energydensity, ~ m |∇ ψ | . Consequently, the composite vortexcores in Fig. 1, where ψ is in M − M q , have diameters L q ∼ s Kβq ∼ ~ q r n βm (10)where the energy scale is taken from Eq. (4) and thecomponent cores, where ψ varies through H − M , havetypical diameters a c ∼ q Kn β .The more complete version of the equipartition ar-gument, given in Sec. III B, estimates the total en-ergy of the vortex as follows: the kinetic energy out-side the composite core is about K l π ln RL , accordingto Eq. (9), the magnetic field energy inside the core isabout ǫ ( q ) L . Minimizing the sum, which has the form − c K ln L + ǫ ( q ) L + c , gives L ∼ q Kǫ ( q ) . The analogous argument also applies to vortices in a single-componentcondensate, giving the core size q Kn α . Because the com-posite cores in a cyclic condensate have some substruc-ture (the component vortices), the actual coefficient c of the logarithm for the cyclic condensate will turn outto be less than l π once the kinetic energy of the vorticesinside the core is included (see Section II B).If q is very small, the component cores are muchsmaller than the composite vortices, and may be regardedas points. The wave function can therefore be approxi-mated by a field that always stays in M except at “sin-gularities” corresponding to the component cores: ψ ( x, y ) ≈ e − iα ( x,y ) ˆn ( x,y ) · F e iθ ( x,y ) √ n χ except at “points” , (11)where χ is the spinor defined in Eq. (2). Over distancesmuch greater than L q even the composite cores seem todwindle to points, suggesting the following approxima-tion: ψ ( x, y ) ≈ e − iα ( x,y ) F z e iθ ( x,y ) √ n χ far from clusters . (12)These expressions generalize[44] the “phase-only approx-imation” for ordinary superfluids[43]; for example, Eq.(11) parameterizes elements of the space M in termsof the symmetries of the q = 0 Hamiltonian, rotations ψ → e − iα F · ˆ n ψ , and rephasings, ψ → e iθ ψ , which are ap-plied to a representative state. Vortices are points whichlook like singularities at the level of resolution exposed byone of these approximations, although what one identifiesas vortex singularity depends on which level of resolutionone uses!Composite vortices result from the order parameterbeing forced out of the minimum-energy space M q bytopology: If the order parameter goes around a hole in M q on some circle in the condensate, then inside the cir-cle, the order parameter has to leave M q . If Eq. (12)continued to hold all the way to the center of the circle,then there would be a singularity since the angular vari-ables α and θ would run rapidly through multiples of 2 π .So we can think of the circle as the core of an extendedvortex in the field in Eq. (12). The singularity is actu-ally filled in by a field of tetrahedra with higher-energyorientations, described by Eq. (11), just as the wave func-tion of a vortex in a single-component condensate avoidssingularities by vanishing in the core. In the filled-inregion of freely-oriented tetrahedra, there are also someelementary “point”-vortices, actually spread out over thedistance a c . These result when the order parameter getstangled around holes in M , and then has to move outsideof this space. These cores are smaller than the compos-ite core because of the large energy scale associated with M . The fields of these vortices can be described qualita-tively by setting q = 0 since kinetic energy surpasses theanisotropy energy very close to the cores.The cores of the component vortices are filled to al-most the same density n as the rest of the condensate(the spin-independent repulsion α favors uniformity since α ≫ β ). Vortices with this property are often referredto as “coreless,” but we use the word “core” in a moregeneral way, so that every vortex can have one. Insteadof defining core as a region where the density vanishes,our definition identifies a core as a region where the wavefunction departs from any particular form. (In the coreof a “coreless” vortex in a ferromagnetic condensate, theorder parameter becomes polar rather than ferromag-netic.) The core of the component tetrahedral vorticesis the region where the order parameter leaves M . Us-ing the absolute ground state space M q leads to a dif-ferent definition of vortex cores of field-aligned vortices:the core is where Eq. (12) breaks down. This is whywe call the region surrounding the tetrahedral vortices a“composite core.” The ordinary definition of a core ariseswhen we focus on another space, S , the sphere definedby | ψ | = √ n . (This space minimizes the largest of theinteraction terms, αn − µn .) From the perspective of S , a core would be a region where the wave function van-ishes to avoid being discontinuous; however, in a spinorcondensate with α ≫ β , the wave function never has tovanish because a closed loop cannot get snagged in thesurface of the simply-connected sphere S .Now we have to classify both the topologies of thepoint vortices (or “tetrahedral charges”) and the com-posite vortices (or “aligned charges”), using the methoddescribed e.g. in [10]. For the point vortices we must pa-rameterize the symmetry group that generates the com-posite core spin textures by a simply connected group G ∗ .We will take G ∗ = { ( g, θ ) | g ∈ SU and θ ∈ R } wheremultiplication is defined by ( g , θ )( g , θ ) = ( g g , θ + θ ). We parameterize G ∗ as follows, D ( e − iα ˆ n · σ , θ ) = e iθ − iα ˆn · F , (13)allowing us to regard G ∗ as a (redundant) description ofthe symmetries of the Hamiltonian. Using this represen-tation, one can define the net vortex charge inside anyclosed loop λ (see Fig. 5a); one simply expresses the wavefunction along the loop in the form ψ ( φ ) = D ( g ( φ ) , θ ( φ )) √ n χ (14)where φ parameterizes the loop (0 < φ < π ). In orderfor Eq. (14) to be continuous when the circuit is closed, ψ ( φ = 0) = ψ ( φ = 2 π ), or D ( g (0) , θ (0)) χ = D ( g (2 π ) , θ (2 π )) χ . (15)It follows that the “classifying group element” or net“topological charge” inside the curveΓ( λ ) = ( e − iα λ ˆ nλ · σ , θ λ ) ≡ ( g (0) − g (2 π ) , θ (2 π ) − θ (0))(16)is a symmetry describing the net rotation and rephas-ing around the closed loop. (Let us also use thebriefer notation g λ = e − iα λ ˆ nλ · σ for the net rota-tion.) Since the tetrahedron has only 24 symmetries (see Sec. I B), this makes for a tractable classifica-tion of the topologies. These topologies form the groupof tetrahedral charges, multiplied together according to( g , θ )( g , θ ) = ( g g , θ + θ ).When q is very large then Eq. (12) has to apply ba-sically everywhere and when q is small then Eq. (12)has to apply outside of bound clusters of vortices. Inorder to classify the topology of vortices in the formercase or of vortex clusters in the latter, we need a sim-ply connected group that parameterizes the q = 0 sym-metry group e iθ e − iαF z ; we take G ∗ q = { ( α, θ ) | α, θ ∈ R } and use the mapping D ( α, θ ) = e iθ − iαF z . Then we write ψ ( φ ) = D q ( α ( φ ) , θ ( φ )) √ n χ and define Q ( λ ) = ( α λ , θ λ ) = ( α (2 π ) − α (0) , θ (2 π ) − θ (0)) (17)where the subscript 3 indicates that the magnetic field fa-vors the 3-fold symmetric orientation illustrated in Fig.2a (since we are assuming c > α λ is understood to be around the field axis.The possible values for Q can be referred to as “field-aligned charges” since they describe the topologies ofvortex-fields in which the tetrahedra keep the orientationfavored by the magnetic field.Now let us consider the form of the fields near a tetra-hedral vortex core. Near vortex i , the spin texture willbe rotationally symmetric, and given by ψ = e i θi π φ e − i αi π φ ˆ n ′ i · F √ n χ (18)where φ is now the azimuthal angle φ centered at thisvortex. Because the tetrahedra near this vortex may betilted, we use √ n χ , a generic member of the cyclicorder parameter space. The vortex is azimuthally sym-metric since the parameters θ i , α i , ˆ n ′ i are constants. Therotation axis ˆ n ′ i is a local symmetry axis for the possibly tilted tetrahedra. To reduce the possible symmetries to afinite set, we should relate the spinor χ to the spinor χ corresponding to the tetrahedron as oriented in Fig. 2a.If we write χ = D ( R, ξ ) χ for an appropriate rotation R and phase ξ , the vortex in Eq. (18) can be written ψ ( φ ) = D ( R, ξ ) e i θi π φ e − i αi π φ ˆ n i · F χ , (19)where ˆ n i = R − ( ˆ n ′ i ) (20)and we have used the transformation rule for angularmomentum: D ( R, ξ ) † F i D ( R, ξ ) = X j =1 R ij F j . (21)(The right-hand side uses the SO matrix R ij associ-ated with the rotation R .) Continuity implies that therotation axis ˆ n i is one of the finitely many symmetryaxes illustrated in Fig. 2a; furthermore, according to theabove scheme, the group element that classifies this vor-tex is Γ i = ( e − iα i σ · ˆ ni , e iθ i ). There are only a discreteset of possible charges when we use ˆ n i , the axis relativeto the body axes of the tetrahedron rather than ˆ n ′ i , theaxis relative to the lab coordinates.Eq. (19) expresses the vortex as a product of a con-stant matrix D ( R, ξ ) (the phase ξ is unimportant) anda standardized vortex configuration. The transformation D ( R, ξ ) rotates the standardized configuration in spinspace, changing both the rotation axis (from ˆ n i to thelocal axis ˆ n ′ i ) and the orientation of the tetrahedra.In the distant surroundings of a cluster of vortices,the spin texture will again have a rotationally symmetricform. The group element describing the change in theorder parameter as one tours the loop λ enclosing theentire cluster is given byΓ( λ ) = Y i Γ i . (22)This algebraic law has a few consequences. First, it leadsto a conservation law that constrains vortex alchemy: thenet charge Γ( λ ) has to be conserved as the vortices insidethe loop combine and metamorphose, since the topologyon the loop cannot change suddenly. (There is only adiscrete set of vortex charges because charges are definedusing the body coordinates.) Second, Eq. (22) restrictsthe types of vortices which can form a cluster when q = 0.(See Fig. 5b.) Outside the cluster, the tetrahedra mustbe aligned with the field (see Eq. (12), so the chargeis described by a field-aligned order parameter Q ( λ ) =( α, θ ) . Eq. (22) requires that Y i g i = e − iα σz (23)and P i θ i ≡ θ (mod 2 π ). The group elements must multi-ply to form a rotation about the z -axis to avoid the largeZeeman cost outside the clusters, or “composite cores”of the vortex molecules.Let us now review the example in the previous sec-tion: the vortex molecule was made out of two vor-tices of type Γ = ( e − iπ √ ( √ σ x + σ z ) , =( e − iπ √ σx + σz √ ,
0) = ( − id,
0) = ( e − iπσ z , Q = (2 π, . This uses the fact that, in SU , all 2 π rotations are equal to − id , where id is the2 × B. Notation for Vortices with and without aMagnetic Field
Let us first assign names to the tetrahedral chargescorresponding to closed loops in M . At q = 0 or withinvortex clusters where q can be neglected, the topologicalcharges are described by a pair Γ = ( g, θ ) (defined in Eq.(16)) where g = e − iα ˆn · σ ∈ SU and θ ∈ R . Note that for the phase of the order parameter to be continuous, asEq. (15) requires, the allowed values of θ and g must becorrelated[35, 37], so only a discrete sequence of phasesmay accompany a given rotational symmetry. Also notethat in SU , rotation angles are defined modulo 4 π ratherthan 2 π . The twelve symmetries of the tetrahedron ac-cording to the ordinary method of counting become 24because, e.g., a clockwise 240 ◦ rotation around an axisis distinguished from the counterclockwise 120 ◦ rotation.This is more than a technical point: the vortex where theorder parameter rotates through − ◦ about an orderthree axis cannot deform continuously into one where theorder parameter rotates through 120 ◦ , and it has moreenergy as well. On the other hand, the α = 4 π “vortex”can relax continuously to a state free of vortices. The ne-cessity of using SU instead of SO is the biggest surpriseto come out of the topological theory.Since g must be a symmetry of the tetrahedron cor-responding to √ n χ , illustrated in Fig. 2a, we candescribe g by indicating its rotation axis using the la-bels from the figure. We refer to the minimal rotationaround a given axis using just the label of the axis;hence S, P, Q, R refer to the rotations through 120 ◦ coun-terclockwise as viewed from the tips of the correspond-ing arrows, and A, B, C refer to counterclockwise rota-tions through 180 ◦ , about A, B, C . Rotations throughlarger angles can be written as powers of these rotations.Therefore P is a 240 ◦ rotation; also P = A = − id since 360 ◦ rotations around any axis correspond to − id in SU . We find it convenient to describe each rotationas a rotation through an angle α around some axis, where − π ≤ α ≤ π . (Positive and negative α ’s correspondto counterclockwise and clockwise rotations respectively.)An arbitrary rotation angle can be replaced by an angle inthis interval using the fact that 720 ◦ rotations in SU (2)are equivalent to the identity. For example, the 480 ◦ counterclockwise rotation P is the same as the − ◦ clockwise rotation P − because P , a rotation throughtwo full turns, corresponds to the identity of SU (2). Onthe other hand, the 240 ◦ counterclockwise rotation P isnot equivalent to the clockwise 120 ◦ rotation around thesame axis, since the corresponding SU (2) matrices differby a minus sign.Now we can list all the pairs of rotations and phaseswhich are allowed by the continuity condition, Eq. (15).The possible vortices according to Ref. [35, 37] are( R m , πm + 2 πn ) and ( A, πm ) where n and m are in-tegers, as well as the corresponding vortices with R re-placed by P , Q , or S and A replaced by B or C .One can work out explicit expressions for the SU (2)elements corresponding to given rotations. For example,let us find the SU (2) element corresponding to A ; sincethe rotation angle is 180 ◦ , A = e − iπ ˆ a · σ = − i ˆ a · σ , (24)where ˆ a is the A -axis. Note that the tetrahedronhas its vertices at (0 , , − − √ , , ), ( √ , q , ),( √ , − q , ). The A axis bisects the segment connect-ing the last pair of points; the midpoint of these twopoints is 12 ( ˆ p + ˆ q ) = ( √ , ,
13 ) . (25)The unit vector ˆ a is obtained by normalizing this vector,so ˆ a = ( r , , r
13 ) . (26)Hence A = − i r σ x − i r σ z . (27)The net charge, Eq. (22), of a set of vortices resultsfrom multiplying the matrices g for the vortices of theset. The result can be identified as one of the rota-tions A n , B n , P n , etc. This procedure completely deter-mines the SU (2) product element, whereas the geometricmethod of applying the appropriate sequence of rotationsto a tetrahedron does not determine the sign of the SU (2)matrix.We will describe a vortex reaction with the followingnotation,( − id, → ( R, π ∗ ( R, π ∗ ( R, − π . (28)Each factor describes the charge of an individual vortex- atom , rather than a cluster of atoms, although ( g, θ ) canbe used to describe the net charge of a set of vortices aswell, as in Fig. 5a. This reaction describes onne point-vortex breaks up into three vortices. The ∗ is just a sepa-rator between the different reaction products, remindingone to check conservation of charge by multiplying bothsides of the reaction out.Another useful cyclic spinor is χ = − i √ . (29) This spinor corresponds to the tetrahedron in Fig.2b. Its vertices are at the points of the form( ± √ , ± √ , ± √ ) if we restrict the choices of signs sothat there are always 0 or 2 minus signs. The fact thatthe A , B , and C axes of this tetrahedron correspondto the ˆ x , ˆ y and ˆ z coordinate vectors makes the spinor √ n χ especially convenient for determining the conse-quences of the quadratic Zeeman term in the next sec-tion. The orientation of the tetrahedron in Fig. 2bis also very convenient for working out the group ofcharges, since the expressions for the symmetry axes are so simple. (E.g., ˆ p = ( √ , − √ , − √ ), since ver-tex P is in the x > , y < , z < P = e − i π ˆ p · σ = − iσ x + iσ y + iσ z .)At nonzero magnetic field, only rotations around the z -axis are symmetries. When c >
4, the ground state spaceis M q , consisting of rotations and rephasings of √ n χ ,as in Eq. (12). Vortices are described as in Eq. (17)by an ordered pair ( α, θ ) describing the rotation andrephasing angles of the vortex. The subscript 3 is usedto indicate that the z -axis is an order three symmetry ofthe c > Q = ( α, θ ) = ( 2 πm , π ( m n )) . (30)When c <
4, minimizing Eq. (4) implies that themagnetic field axis is an order two symmetry, and theground state space is M q , the rotations and rephasingsof √ n χ . Now ( α, θ ) specifies the vortex types. Thepossibilities are Q = ( α, θ ) = ( πm, πn ) . (31) II. ENERGIES AND SYMMETRIES
This section considers the effect of the magnetic field,which binds vortices, and the kinetic energy, which keepsthe bound vortices from merging altogether. These areboth included in the full energy function H = Z Z d r ~ m ∇ ψ † ∇ ψ + V tot ( ψ ) , (32)where V tot is given by Eq. (3). When B = 0, the Zee-man effect introduces an extra phase boundary dividingthe cyclic phase into two phases, with a phase transitionat c = 4 where the tetrahedron changes its orientationrelative to the magnetic field. (This result applies formoderate magnetic fields; very low magnetic fields causedifferent transitions[26].) A. The Anisotropy potential from the QuadraticZeeman effect
Let us determine which orientations of the tetrahedronwill be preferred by a magnetic field along the z axis. Thepreferred orientation can be calculated for small q froman effective potential which is a function of the orienta-tion of the tetrahedron. As long as q ≪ n β, (33)the tetrahedron will be only slightly deformed. It willmove into a space M ′ displaced by a distance on theorder of qn β from the space M of arbitrarily oriented0perfect tetrahedra. The spinors in the distorted spaceare given by ψ = √ n D ( R, ξ ) χ + δψ, (34)where D ( R, ξ ) is the spin two rotation matrix correspond-ing to the rotation R of space, multiplied by a phase. Thedistortion δψ depends on the orientation R . Eq. (11),which omits the deformation, is a harmless shorthanddescription, emphasizing the orientation of the tetrahe-dron. In this section, we use √ n χ as the standardspinor orientation instead of √ n χ to simplify calculat-ing the energy; conveniently, the body axes A, B, C ofthe corresponding tetrahedron for the former state arealigned with the coordinate axes ˆx , ˆy , ˆz . The body axesof the rotated state D ( R ) √ n χ (which make the an-gles α , α , α with the z -axis) are thus R ( ˆx ), R ( ˆy ), R ( ˆz ).Therefore the z component of the spin, in terms of thecomponents of the spin along the body axes, is D ( R ) † F z D ( R ) = cos α F x + cos α F y + cos α F z . (35)At first order the quadratic Zeeman effect does nothave any dependence on the orientation of the tetrahe-dron because a tetrahedral spinor is “pseudoisotropic,”i.e., χ † F i F j χ = 2 δ ij . The first order energy is thusgiven by < qF z > ≈ n qχ † D ( R ) † F z D ( R ) χ ≈ n q X i,j =1 cos α i cos α j χ † F i F j χ ≈ n q, (36)which does not prefer any orientation of the tetrahedron.In the last step we used X i =1 cos α i = 1 (37)which follows from the fact that ˆ z , a unit vector, hasbody-coordinates (cos α , cos α , cos α ).To find the second order energy due to the quadraticZeeman effect, which will break the tie, we have to findthe deformed state and its energy. The deformation δψ isdetermined by minimizing the total interaction and Zee-man energy in Eq. (3) for each given orientation R . Nowif the deformation is not restricted somehow, the “defor-mation” which minimizes the energy will be very large,involving the tetrahedron rotating all the way to the ab-solute ground state.We therefore allow deformations onlyof the form δψ = dD ( R, ξ ) χ + 1 √ aD ( R, ξ ) F x χ + bD ( R, ξ ) F y χ + cD ( R, ξ ) F z χ )+ ( e + if ) D ( R, ξ ) χ t , (38) where a, b, c, d, e, f are real numbers. These terms cor-respond to the excitation modes found by [45]. Thiscorrection only perturbs ψ in 6 of the 10 directions inthe Hilbert space. The other 4 directions are accountedfor by the rotation R and the phase ξ which would beGoldstone modes when q = 0. (The energy remains ξ -independent even when q = 0.) The particular six stiffdeformations in Eq. (38) are chosen because they are or-thogonal to infinitesimal rotations and rephasings of thetetrahedral state. We have to find the deformations thatminimize V tot , Eq. (3), for each rotation R .To evaluate V tot , note that D ( R, ξ ) cancels from all theterms in the energy except for the Zeeman term, whereone can use Eq. (35). The resulting expression for theenergy density reads V tot ( ψ ) = 12 α ( ˜ ψ † ˜ ψ ) + 12 β ( ˜ ψ † F ˜ ψ ) + 12 γ | ˜ ψ † t ˜ ψ | − µ ˜ ψ † ˜ ψ − q X i,j =1 cos α i cos α j ˜ ψ † F i F j ˜ ψ (39)where ˜ ψ = √ n χ + dχ + √ ( aF x χ + bF y χ + cF z χ )] +( e + if ) χ t is the perturbed wave function without therotation. The effective potential Eq. (4) is obtainedby minimizing V tot over a, b, . . . while keeping R fixed.Further details are in Appendix A. (Working with ˜ ψ is equivalent to fixing the orientation of the tetrahedronand rotating the magnetic field.)The effective potential suggests an analogue of themagnetic and charge healing length in condensates with-out magnetic fields (the “tetrahedron tipping length”)–when the tetrahedra are rotated out of the appropriateground state orientation at the edge of a condensate, theenergy in Eq. (4) returns the order parameter to M q within the distance L q ∼ ~ q q n βm .Now the ground states can be found as a function of c . When c >
4, a short calculation shows that Eq. (4)has its minimum at cos α i = ± √ , i = 1 , ,
3; i.e., whenthe magnetic field is along the line connecting a vertexof the tetrahedron to the opposite face or vertex. Hencethe order parameter space M q is as given in Eq. (12).When c <
4, the effective potential is minimized by anorientation in which the field points parallel to the linejoining a pair of opposite edges (see Fig. 2)[54].The dependence of the ground-state orientation on c can be understood intuitively using the geometricalrepresentation of Ref. [36]. In a tetrahedral state, m = ψ † F ψ = 0 and θ = ψ † t ψ = 0, minimizing the in-teraction energies in Eq. (3). When a magnetic field isapplied, the base of the tetrahedron illustrated in Fig. 2agets pushed toward the vertex at − ˆ z . The tetrahedronin Fig. 2b, on the other hand, has its upper and loweredges pushed together, toward the xy -plane. Both thesedeformations move the spin roots (see [36]) away fromthe north and south poles, which increases the probabil-ity that F z = ± m of the spinorbecomes nonzero, increasing the β term of Eq. (3). In thesecond case, the magnetization of the spinor is still zero,by symmetry, but one can check that the spinor is nolonger orthogonal to its time-reversal, so θ is nonzero[55].Therefore the orientation of the ground state tetrahedronis determined by whether the m term or the | θ | has alarger coefficient in the Hamiltonian. If the coefficient ofthe magnetization term is very large, then the state with-out any magnetization is preferred. The detailed calcu-lation shows that the relevant comparison is between cβ and 4 β .The main source of anisotropy is different at suffi-ciently low magnetic fields[26]; the cubic Zeeman effect,proportional to B , then dominates over the effectivepotential in Eq. (4), proportional to B . Neverthelessthe the order B effect we have calculated can dominateover the B effect, even when the magnetic field is smallenough for the perturbation theory just described to ap-ply. This is possible because the denominator n β in Eq.(4) is small compared to the hyperfine energy splitting A HF . For spin 2 atoms the effect of the magnetic field isgiven by V nz = ψ † q ( µ B B ) + A HF + A HF µ B BF z ψ (40)where A HF is the hyperfine coupling. It follows that thequadratic and cubic Zeeman effects are qF z = µ B B A F z and µ B B A F z . The analysis we have given applies whenthe magnetic field is weak enough that the wave functionis not drastically distorted, Eq. (33), but strong enoughfor the second order effect of the quadratic Zeeman termto dominate over the cubic Zeeman term. These condi-tions combine into n β << µ B B << p A HF n β (41)For Rb at density 5 × /m , n β = 3 nK and A HF =160 mK while µ B = 67 µ K/Gauss; hence the anisotropypotential used here is actually valid for a wide range ofmagnetic fields, between .
04 mG and . B. Kinetic Energy
Now that we have estimated the energy due to mis-alignments with the quadratic Zeeman field, let us de-termine the kinetic energy of vortices in terms of therotations and rephasings, Eq. (16). This energy will bethe source of the repulsion that keeps the vortices apartwithin the vortex molecules. The kinetic energy is de-fined by the gradient terms in the Hamiltonian H ; theanalogue for liquid crystals is the elastic energy favoringalignment of the order parameter.In order for the kinetic energy of a vortex to be min-imal, the field on a circle around it should trace outa geodesic, as mentioned above. For the cyclic phase, geodesics take the form given by Eq. (18). To see thatvortices are geodesics as functions of φ , suppose the fieldof a vortex is given far away by the radius independentexpression ψ ( r, φ ) = √ n F ( φ ) , (42)for an appropriate spinor function F ( φ ). The main con-tribution to the energy of this vortex is from the kineticenergy, which can be estimated by integrating from thecore radius a c ∼ ~ q πn βm . This is the same as the spinhealing length (see [16] for the definition). The kineticenergy is E ≈ Z Z d r ~ m ∇ ψ † · ∇ ψ ≈ Z π n | F ′ ( φ ) | dφ Z Ra c ~ mr rdr ≈ ~ n m ln Ra c Z π | F ′ ( φ ) | dφ (43)Now the curve parameterized by F ( φ ) adjusts itself soas to minimize the last integral, while maintaining thetopology of the circuit traced out by F ( φ ) in the orderparameter space. One can show that an integral of thisform is minimized when F ( φ ) traces out a closed geodesic in the ground state space. The length of a closed curve isdefined by R π | F ′ ( φ ) | dφ , so it is not surprising that thegeodesic of charge Γ, which minimizes this expression,also minimizes Eq. (43). Furthermore, if the geodesichas length l Γ , then R π dφ | F ′ ( φ ) | = l π .A geodesic in the cyclic order parameter space (whichhas the local geometry of a perfect sphere in four di-mensions) can be described by a rotation at a fixed ratearound a single axis as in Eq. (18). (For a shape lessisotropic than a tetrahedron, the order parameter wouldrotate around a wobbling axis according to the rigid-rotation equations.) Substituting the symmetrical F ( φ )into Eq. (43) gives[44] E ≈ ~ n m ln Ra c Z dφψ † ( − α ˆn · F π + θ π ) ψ ≈ ~ n πm ln Ra c [ α ( n i n j Q ij + s ( s + 1)3 ) − αθn i M i + θ ] (44)where the general expression for any spin and phase hasbeen given in terms of the quantum fluctuation matrix Q ij = n ( ψ † F i F j + F j F i ψ ) − s ( s +1)3 δ ij and the magnetiza-tion per particle M i = m i n . The spin 2 tetrahedron stateappears to be isotropic as long as one does not go be-yond second order correlators, as seen from the followingcalculations: < χ | F i | χ > = 0 (45) < χ | F i F j | χ > = 2 δ ij (46)2and hence M = Q = 0. Eq. 44 implies that the en-ergy is proportional to l = ( θ + 2 α ), a generalizationof the Pythagorean theorem showing how to combinethe amount of rephasing and rotation to get the totalgeodesic length. A spin rotation costs twice as much en-ergy as a rephasing by the same angle.In order to study vortex stability and Coulomb forces,let us define the “energy index,” I E = ( l Γ π ) = ( θ π ) + 2( α π ) , (47)which is a fraction for each tetrahedral charge from Sec.I B. The energy of a vortex is given as π ~ n m ln Ra c I E (Γ), amultiple of the energy of an ordinary phase vortex. Theforce between a pair of vortices can be expressed verysimply in terms of I E .The force follows from an estimate of the energy ofa cluster of vortices. Using ideas from Ref. [46], wethink of the cluster as forming the “core” of a biggervortex, as illustrated in Fig. 6. (We are not necessarilyassuming that the vortices are bound together.) Drawa circle of radius X just around the group of vortices.The kinetic energy can then be found as the sum of theenergies outside and inside of X ; for the case illustratedin the figure this energy is approximately π n ~ m ([ I E ( X ) ln RL X ] + [( I E (1) + I E (2) + I E (3)) ln L X a c ]) , (48)where L X is the diameter of the group being combinedtogether and R is the radius of whole system. The firstterm describes the energy outside of X . Sufficiently faroutside of X , the field should have the form of a rota-tionally symmetric vortex. The rotation and rephasingof this vortex, measured along a circle outside X , is ob-tained by multiplying the group elements which describeall the individual vortices according to the rule for vortexunification Eq. (22). Hence the energy outside X is givenby an expression like in Eq. (44), except that the integralmust start at the radius L X of the circle, so a c must bereplaced by L X . The energy inside X is approximatedby adding the energies of the three vortices in it, whichare calculated like in Eq. (44) but now with R replacedby L X .This approximation makes a small error (compared toln L X a c ) by ignoring the region where the vortices’ fieldsoverlap. This error has a special scaling form if q = 0and if there are no other vortices in the condensate. Inthis case, the kinetic energy gives a complete descriptionof the vortices outside their cores, and has a symmetryunder rescaling. Therefore (as in [46]), one can show thatthe difference between Eq. (48) and the actual energy hasthe form ∆ E = πn ~ m f ( L L , L L ) + O ( a c L X ) (49)where L ij refer to the sidelengths of the triangle formedby the vortices. This correction can be ignored relative PSfrag replacements XX FIG. 6: A group of vortices is combined into the core of a com-posite vortex X . The energy outside the blacked-out “core”is calculated from the resultant winding number of the vor-tices inside, and the energy inside the “core” is calculatedby looking inside the core to see which vortices are actuallythere. to the logarithmically divergent terms kept in Eq. (48), as long as the side-lengths all have the same order ofmagnitude , since in this case f has no singularity.The general expression for the energy of a set of vor-tices with charges Γ i , after rearranging the formula toemphasize the dependence on the diameter of the set L ,is: E K ≈ ~ πn m [ X i I E (Γ i ) − I E ( Y i Γ i )] ln La c + ~ πn m I E ( Y i Γ i ) ln Ra c (50)As long as the vortex cores are well-separated, the errorin this estimate depends only on the ratios of the dis-tances between the vortices, as in the three-vortex case,Eq. (49). The energy function has a similar form tothe two-dimensional Coulomb interaction between vor-tices in an ordinary single-component condensate, andshows that the vortices repel or attract each other ac-cording as the index of the combined vortex is greater orless than the sum of the separate vortices’ indices. Aninteresting difference is that the interaction is not a sumof interactions of all pairs (consider three ( A,
0) vortices,for example, or almost any other set of three vortices).When there are just two vortices, the force betweenthem can be calculated by differentiating Eq. (50), giv-ing − ~ πn mL [ I E (Γ ) + I E (Γ ) − I E (Γ Γ )]. This is the exact (if L ≫ a c and q = 0) expression for the force be-tween the two vortices because the error (see Eq. (49))reduces for two vortices to a constant. Improving on Eq.(50) depends on finding the spin texture in the overlapregions, and these equations are nonlinear when not allthe vortices use the same symmetry axes.In the next section, we will apply Eq. (50) morebroadly. When there is a small cluster of vortices of size L , but there are some other vortices besides, the energyof just the vortices in the cluster can be estimated byreplacing R in Eq. (50) by the distance D to the nearestvortex not in the cluster. Likewise, for q = 0 the energyof a cluster with a net charge that is field-aligned can be3estimated by replacing R by L q . As long as L ≪ L q ,the detailed expression for the energy has the form givenby Eq. (49), but when L approaches L q , the anisotropyenergy starts to compete with the kinetic energy and todistort the spin texture around the vortices. III. CHEMISTRY OF VORTICES
In this section we will first discuss stability of isolatedtetrahedral vortices and then determine when these vor-tices can combine to form molecules (and what the spintexture around a molecule looks like). Some of thesemolecules are only metastable and can each break up inseveral ways.Bound states of vortices will be formed out of stabletetrahedral vortices. These are the vortices based on 120 ◦ symmetries of a tetrahedron, accompanied by a phaseshift of π or − π , and vortices based on 180 ◦ symmetrieswithout a phase shift. Vortices with larger rotations orphase shifts will not occur as components of molecules.The net charge of the bound state must be field-aligned,with a schematic relation Q = Y i Γ i (51)between the field-aligned charge and the componentcharges. A composite vortex is stable if Coulomb repul-sions can prevent the collapse of any subset of its com-ponents; since the Coulomb interaction is not a sum ofpairwise interactions, it is not enough that every pair ofvortices repel each other. Finally any bound state whosecharge Q = ( α, θ ) has big enough rotation or rephasingangles can in principle decay into molecules with less en-ergy, but this can only occur if thermal energy overcomesthe Coulomb repulsion between the component vortices.The following expands on this general picture and pointsout some interesting sidelights. A. Stable Tetrahedral Vortices
Only stable vortices will be found in the core-regionof composite vortices. Since a weak anisotropy term canbe neglected near component vortices, we can enumeratethe stable vortex types assuming that q = 0.Absolute stability of a vortex with charge Γ impliesthat if Γ is forced to break up into the fragments Γ i ,then the energy of the fragments grows as they separatefrom one another. The energy of the fragments can befound by applying Eq. (50). Note that Q i Γ i = Γ byconservation of charge. If Γ has phase and rotation angles θ and α then E fragments ≈ n ~ πm [ X i ( θ i +2 α i ) − ( θ +2 α )] ln La + cnst. (52) Thus, if P i ( θ i + 2 α i ) < ( θ + 2 α ), the energy decreasesas the vortices move apart, so the fragments will moveapart by themselves the rest of the way. On the otherhand, the vortex is absolutely stable if X i ( θ i π ) + 2( α i π ) > ( θ π ) + 2 α π for every set of Γ i = ( e − i αiσ · ˆn i , e iθ i ) such that Y i Γ i = Γ(53)Any tetrahedral vortex not satisfying this “absolutestability criterion” can break up into a lower-energystate, and we will assume that this break-up happensspontaneously for these component vortices. For ex-ample, the vortex ( id, π ) should break up into two( id, π )’s, halving the energy. (In fact, these two singlyquantized vortex can break up even further.) Generaliz-ing this example, any vortex with charge ( g, θ ) whose cir-culation θ is bigger than 2 π can break up into ( g, θ − π )and (0 , π ) since θ > (2 π ) + ( θ − π ) . This leavesonly finitely many vortices that have the possibility ofbeing stable: all the ones with phase winding numbersnot more than 2 π . Some of the vortices with | θ | ≤ π arealso unstable; trial and error finds decay processes suchas: 1. ( − id, → ( C, ∗ ( C,
0) or ( R, π/ ∗ ( R, π/ ∗ ( R, − π/ − id, π ) → ( R, π ) ∗ ( R, π ) ∗ ( R, π )3. ( R , π/
3) can break up into ( R, π/ ∗ ( R, π/ S − , π/ ∗ ( C, R , − π/
3) might break up into ( S − , − π ) ∗ ( C, C, π ) could break up into ( R, π ) ∗ ( S − , π ).All of these decays lower the energy index. There arealso two vortices which can break up without the energyindices changing:1. (0 , π ) has the same charge as ( R, π ) ∗ ( R − , π ).2. ( R, − π ) has the same charge as ( P − , − π ) ∗ ( Q − , − π ).The energy may either increase or decrease after one ofthese processes, the logarithmic term which dominatesthe energy (see Eq. (50)) does not change, so the remain-der term needs to be calculated to determine whetherthese break-ups raise or lower the energy. A point vortexwhose charge is (0 , π ) is probably unstable if α ≫ β because a pure phase vortex has to have an empty core(with energy density of order αn ), while the two frag-ments it breaks up into just have non-cyclic cores (energyof order βn ). [56]We can now enumerate the vortices which are stable,noting that some vortices (such as ( P , π )) can break up4similarly to the ones just listed on account of symmetry.The only vortex types that have not been eliminated arethe vortex with one-third circulation, ( R, π/
3) (energyindex ) and the currentless vortex ( C,
0) (energy index ) as well as possibly ( R, − π ) (energy index ), andalso the inverses and conjugates of these. B. Vortex Molecules at q = 0 and their SpinTextures In this section we will describe a qualitative wave func-tion for the example in Section I A to illustrate how thefield of a vortex molecule deforms in response to theanisotropy energy as one leaves the region containing thetwo vortices. We will then give the binding criteria whichdescribe how to use group theory to check which sets ofthe stable tetrahedral vortices (from the previous sec-tion) can combine together to form a vortex molecule ata nonzero magnetic field.A vortex with a charge that is not compatible with themagnetic field has an energy that grows proportionallyto the area of the condensate. Eq. (23) shows that thevortices in a cluster can avoid this energy cost if they havea net charge corresponding to a rotation around the z axis. More specifically, a set of tetrahedral vortices, withtopologies Γ i , has to have a net topology of the form Y i Γ i = ( R m , πm πn ) . (54)The “molecule” can then have the rotational and phasewindings ( α, θ ) = ( πm + 4 πj, π ( m + n )) at infin-ity, for any j . One example of a molecule for the M q phase is the composite vortex discussed at the beginning,( A, A, Q = (2 π, in the field-aligned condensate.Writing a wave function that describes this exampleeven qualitatively is a little more complicated than itsounded in Sec. I A. A possible wave function is illus-trated in Fig. 7. We can build this wave function upin stages. The most obvious attempt at writing a wavefunction fails to eliminate the quadratic energy cost: ψ misalign = e − i ( φ + φ )( √ F x + √ F z ) √ n χ , (55)where φ i = arctan y − y i x − x i is the polar angle measured withrespect to the location ( x i , y i ) = ± ( L ,
0) of the i th vor-tex. The two 180 ◦ rotations about A combine into a 2 π rotation on a circle surrounding both vortices, but notabout the magnetic field axis. This wave function takesthe form ψ ≈ e − iφ √ ( √ F x + F z ) √ n χ at infinity where φ and φ approach φ . The tetrahedra rotate arounda tilted axis, so they do not stay aligned appropriatelywith the magnetic field except on the x -axis and the y -axis (where the tetrahedra are reversed, but this is stilla ground-state). The Zeeman energy of ψ misalign is stillinfinite. This wave function agrees pretty well with the one illustrated in the figure within the region surroundedby the large circle. The vortex cores are cordoned offby the small circles, where the 180 ◦ symmetry axes aremarked by dots. Note that the tetrahedra three rows in-side the large circle behave like the tetrahedra at infinitydescribed by Eq. (55); they rotate through 360 ◦ aboutthe order-two axis bisecting their right edge, producingall sorts of arbitrary orientations.Another attempt, which uses the R axis instead of the A axis to eliminate the tilting of the tetrahedra, ψ discontinuous = e − i ( φ + φ ) F z √ n χ , (56)is a complete fiasco, since this function is not continuousalong the line connecting the two cores. (The R axis hasthe wrong symmetry, so as ( x, y ) circles around ( x , y ), ψ discontinuous changes from χ to e − iπF z χ = χ .)Luckily, since a 2 π rotation around one axis can bedeformed to any other (2 π rotations all correspond to − id ∈ SU ) one can produce a spin texture which has afinite Zeeman energy and is continuous. A hybrid of Eqs.(55) and (56) which achieves this, illustrated in Fig. 7,is: ψ = e − iφ ( F x sin µ ( r )+ F z cos µ ( r )) e iφ √ Fx + Fz √ × e − i ( φ + φ )( √ F x + √ F z ) √ n χ (57)where µ ( r ) must satisfy µ (0) = arccos r
13 (58) µ ( ∞ ) = 0 (59)For example, we could definecos µ ( r ) = r r + D r + 3 D (60)where D is a variational parameter. (Probably D ∼ L isoptimal.)This field has two vortices at ( ± D , ˆ n ′ axis which is aligned with the symme-try axis of the tilted tetrahedra around the vortex; thetilting is carried out by the rotations described by thefirst two factors in Eq. (57), just as discussed in SectionI A. Near vortex 1, the wave function is approximately ψ ≈ e − iφ ( √ F x + √ F z ) √ n χ (61)and near vortex 2, ψ ≈ e − iπ ( F x sin µ ( L )+ F z cos µ ( L )) e − iφ iπ ( √ F x + √ F z ) √ n χ . (62)5(The local symmetry axis ˆ n ′ is the rotation of thestandard axis q ˆ x + q ˆ y , through a half-turn about ˆ x sin µ ( L ) + ˆ y cos µ ( L ), according to Eq. (18).)Though the two tetrahedra are rotated into misalignedpositions inside the large circle, the excursions from M q to M are brief and therefore cost only a finite amountof energy. Indeed, the first two factors fix the field up atinfinity by applying a continuously varying rotation tothe overall texture. These factors change the axis from A to R at large r as one can see by replacing φ ≈ φ by φ and using Eq. (59): ψ → e − iφF z √ n χ . (63)The 360 ◦ axis changes from the A axis to the R axisas one crosses through the transition region indicated bythe large circle in Fig. 7. As you follow a radius out-ward past the circle, the tetrahedra are tipped by dif-ferent amounts. (The tetrahedra on the negative x -axishave to be tipped the most, though their original ori-entation is compatible with the magnetic field! The facewhich is on top is changed. This reorientation is requiredto make the amount of tipping continuous: the tippingangle increases more and more as φ goes from 0 to π .)The first two factors manage to fix the orientation atinfinity without introducing discontinuities like in Eq.(56). Although they seem to have a vortex-like discon-tinuity at the origin, their discontinuous parts cancel onaccount of Eq. (58). (Hence the tetrahedra near the ori-gin in Fig. 7 all have roughly the same orientation.) Inshort, though ( A, = ( − id, can be deformed as r → ∞ so that the Zeeman energy issmall.The kinetic energy of this composite vortex, accordingto Eq. (50), is E K ≈ ~ πn m [2 I E (( A, − I E (( − id, La c + π ~ n m I E (( − id, Ra c . (64)Since I E (( A, and I E ( − id,
0) = 2, the coefficientof the first term is negative, so the vortices are drivenapart in order to decrease the kinetic energy. This effectcompetes with the anisotropy energy, which tries to bringthe vortices together. The net energy is therefore E ≈ f ( D/L ) q D β + π n ~ m (2 ln RL + ln La ) . (65)This is a more complete version of Eq. (8). f summarizesthe dependence of the anisotropy energy Eq. (4) on D and L , but is too complicated for us to figure out!The competition between the attractive Zeeman-termand the repulsive kinetic-energy term determines the size of the vortex. Assuming D = L and minimizing over L gives L ∼ L q = q n ~ βm q . (66)This composite core-size is also the scale for the de-cay of tipping of tetrahedra due to the competition ofanisotropy and kinetic energy (see Sec. II A), just as thecore size of a spin vortex in a spin 1 condensate is equalto the magnetic healing length. Vortex molecules willhave a size of the same order of magnitude (even if thereare more than 2 subvortices) except when the componentvortices have a short-range repulsion, like the two exam-ples at the end of Section III A. (Such components formsmaller molecules; vortices never form larger molecules.If a pair of tetrahedral vortices is stretched beyond L q , a“cord” of tipped-tetrahedra forms between them, a sim-ple version of the string imagined to hold the quarks to-gether in a rapidly-rotating baryon.)To find the total energy of the molecule, substitute L into Eq. (65) and simplify using a c = ~ q πn βm : E vortex ∼ π n ~ m ln R √ mq ~ (67)We have dropped the contribution from the quadraticZeeman term since it adds something that is independentof q .This formula can be compared to the energy one ex-pects for a simple vortex with the same charge (2 π, : E vortex = I E ((2 π, ) π n ~ m ln Ra ′ c + ǫ c (68)which includes a core energy ǫ c and allows for a physicaldefinition of the core size. If we take a ′ c to be the size ofthe molecule, L q = q q n ~ βm , then the core energy hasto be ǫ c ≈ π n ~ m ln n βq . (69)The core energy becomes large as q → L q .It makes sense to regard this composite vortex as amolecule because Coulomb repulsion keeps the vorticesin the core separate. The previous discussion can be gen-eralized by listing a set of binding criteria; these ensurethat a set of tetrahedral vortices will form a stable ormetastable composite vortex:1. Each component vortex is one of the stable q = 0vortices from Section III A.2. The kinetic energy is not decreased when any sub-set of the component vortices coalesces into a singlevortex.6 FIG. 7: An illustration of the composite vortex ( A, ∗ ( A, c > ◦ vortices, withsymmetry axes indicated by black dots. The large circle indicates the transition region where the axis of the 360 ◦ rotationchanges relative to the tetrahedra, from the A to the R axis. The figure uses a functional form with a rapid jump for µ ( r )(unlike in Eq. (60)) for simplicity, so that the transition region does not overlap the cores. For the probably more realisticform given by Eq. (60), the tetrahedra are already tipped at the centers of the vortices so that the local rotation axes ˆ n ′ i donot have the standard orientations illustrated in Fig. 2a.
3. There is no way for the component vorticesto form submolecules that can break apart.This would occur if the components could berearranged and then partitioned into r sets { Γ , Γ , . . . , Γ j } , { Γ j +1 , Γ j +2 , . . . , Γ j + j } , . . . , { Γ j + j + ··· + j r − +1 , Γ j + j + ··· + j r − +2 , . . . , Γ j + j + ··· + j r } such that each subset forms a molecule thatis compatible with the magnetic field (i.e.,7 Q j + ··· + j k i = j + ··· + j k − +1 Γ i = ( R mk , π ( m k + n k ))) andsuch that the sum of the energy indices of thesesubmolecules is less than the energy index of theoriginal molecule .The vortex molecule ( A, ∗ ( A,
0) clearly satisfies allthese conditions: Condition 1 is satisfied because ( A, A,
0) is compatiblewith the magnetic field.
C. Metastable Vortices and How They Decay
Not all of the vortex molecules satisfying the three con-ditions above are absolutely stable. The analogue of theabsolute stability condition, Eq. (53), also selects a fi-nite set of aligned vortex types when q = 0, this timefrom among the group elements listed in Eq. (30). Theabsolutely stable charges are ± ( 2 π , π , ± ( 2 π , − π . (70)For any other vortex topology Q = ( α, θ ) , one can findvortex topologies Q i such that Q i Q i = Q and I E ( Q ) > X i I E ( Q i ) . (71) Point vortices with such a topology Q would likely breakapart spontaneously.There is actually another pair of charges that could beabsolutely stable, but the energy index estimate is notaccurate enough to decide the issue:(0 , ± π ) , ( ± π , ∓ π (72)These vortices can break up into pairs without changingthe net energy indices, reprising the ambiguous behaviorof the two vortices at the end of Sec. III A. (See SectionIV B for an answer.)There are some composite vortices of other charges be-sides the four listed in Eq. (70) which are long-lived. Theabsolute stability criterion misses this possibility becauseit ignores the details of the vortex cores, drawing all itsconclusions from the topology of the vortices far away:Suppose the initial vortex Q is a cluster of vortices withtopologies Γ , Γ , . . . , Γ r (see Fig. 8). The decay prod-ucts discussed in the previous paragraph, Q i , may be thecombined topology of other vortex clusters. The energyof the vortices after the reaction (Fig. 8b) is smaller if E init = I E ( Q ) π n ~ m ln RL + k ln La c > X i [ I E ( Q i ) π n ~ m ln RL + k i ln La c ] = E fin ; (73) PSfrag replacementsa) b)
Q Q Q Q FIG. 8: Illustration of the absolute stability criterion. Acomposite vortex (a) and a possible set of composite vor-tices (b) it can break up into. Even if Q = Q Q Q and I E ( Q ) > I E ( Q ) + I E ( Q ) + I E ( Q ) so that the energy woulddecrease, the break-up might not occur spontaneously. If thecomponent vortices in (b) are different from the componentvortices in (a), then the vortices making up Q , Q , and Q would have to be produced in a “chemical” reaction from thecomponents of Q . The terms proportional to ln RL stand for the kinetic en-ergies outside the composite cores, and the terms propor-tional to ln La c stand for the energies within the compos-ite cores; the latter contributions do not matter once thecomposite vortices are far apart ( R ≫ L ). Therefore, theenergy-index relation, Eq. (71), implies that the energydecreases when Q → Q ∗ Q ∗ . . . Q r .However, at zero temperature, a vortex molecule sat-isfying the binding criteria cannot break up, even if thetotal energy would end up smaller. Although the meta-morphosis of Fig. 8a into Fig. 8b lowers the energy, theprocess will not occur spontaneously. According to con-dition 3, the vortices in Fig. 8b are different than thosein Fig. 8a. And according to conditions 1 and 2, thereare no spontaneous chemical reactions that can occur tomake the components in Fig. 8b out of those in Fig. 8a.Thus, at zero temperature, composite vortices besidesthe ones with charges listed in Eq. (70), whose energyindices seem to be too big, can still be stable. At nonzero temperature, such a molecule will only be metastable be-cause it can decay after reactions inside its core producethe vortex types that appear in Fig. 8b. These reactionsare prevented by energy barriers, so the decay will occuronly after a long time.The vortex molecule made up of ( A, ∗ ( A,
0) is anexample of a metastable vortex; its charge (2 π, didnot appear in Eq. (70) because it is the same as the netcharge of the three field-aligned point vortices ( π , π ) ∗ ( π , − π ) ∗ ( π , π ) . The energy index of the moleculeis 2(1 ) + 0 (see Eq. (47)) while the energy index of thethree point vortices is smaller, + + . Nevertheless,since these three vortices are not present in the core ofthe original vortex the decay cannot occur spontaneously.A vortex with a composite core can break up by acombination of vortex-fusion and vortex fragmentation.Some subsets of the original component vortices fuse andthe fused vortices each break up into some other vortices.8These regroup into clusters each of which has a chargecompatible with the magnetic field. Then each clustergoes its own way. (The fusion and fragmentation stepsmight sometimes happen more than once.) Conditions1 and 2 ensure that at least one of these steps will beopposed by the Coulomb potential, but the total energywill decrease if the energy index decreases. The vortexmolecule will be long-lived because its components donot know that the hard effort of fusing will allow themto change into vortices which can separate.Trial and error yields a couple of ways in which the( A, ∗ ( A,
0) bound state can break up. One possibilitybegins with the two component vortices coalescing,( A, ∗ ( A, → ( − id, → ( R, π ∗ ( R, π ∗ ( R, − π A, ∗ ( A, → [( R − , − π ∗ ( Q, π ∗ ( A, → ( R − , − π ∗ [( Q, π ∗ ( A, . (75)In the first process, the two vortices come together, in-creasing the kinetic energy in accordance with Condition2 (as shown by calculating the I E ’s and substituting intoEq. (50)). The resulting vortex breaks up into three vor-tices which can separate from each other because theyare compatible with the magnetic field. The increase inenergy during the first stage is given by E s − E init = π n ~ m ln L q a c where E s = π n ~ m I E ( − id,
0) ln Ra c is the en-ergy of the intermediate vortex. Thermal fluctuationshave a chance of driving the ( A, E s − E init .In the second process, Eq. (75), the ( A,
0) vortex firstsplits up into two vortices. The first of these, ( R − , − π )is compatible with the Zeeman field and can leave. Theremaining two vortices form a new molecule which can-not break up because Q is a rotation around the wrongorder 3 axis. For this process Condition 1 requires thatthe energy increases during the initial fragmentation. Tocheck this, note that the energy of the intermediate stateis E s = π n ~ m { [ I E ( R − , − π I E ( Q, π I E ( A, La c + I E ( − id,
0) ln RL } (76)and the energy barrier is E s − E init = π n ~ m [( I E ( R − , − π I E ( Q, π − I E ( A, La = πn ~ m La . (77)This energy barrier is lower than E s − E init , so Eq. (75)is a more common break-up route. (In a finite conden-sate, thermally excited break-ups can be observed only if a vortex molecule is somehow prevented from wander-ing to the boundary of the condensate and annihilatingbefore it can decay.)These two examples illustrate the meaning of the sta-bility conditions. Conditions 1 and 2 ensure that frag-mentation and fusion processes cannot happen sponta-neously. The third condition simply points out thatvortex clusters like ( A, ∗ ( B, will not be stablebecause the components can sort themselves into field-aligned groups and break up without any thermal assis-tance. The second condition can be difficult to checkfor a composite vortex with three or more sub-vortices.One must consider subsets of every size and check thatthey cannot lower their energy by collapsing all at onceinto one vortex. Just knowing that any two vortices ofthe subset repel each other does not guarantee that theset of vortices do not collectively attract each other! Anexample is the set of three vortices ( A, . Any two ofthese vortices would combine to form a vortex with a360 ◦ rotation, ( − id, + , of the indices of the collapsing pair. On theother hand all three vortices could form a vortex ( A − , < I E ( A, q is small, theenergy at the top of the barrier is greater, by a loga-rithmically large amount, than the energy of an initialvariational state like the approximate wave function Eq.(57). A solution to the Gross-Pitaevskii equation shouldresult if one starts from this qualitative texture and letsit relax to a local minimum of the energy. There is notenough energy for the wave function to get over the en-ergy barrier, so the wave function should get stuck in alocal minimum. (The energy of the intermediate state isnot known precisely because of the rough estimates wehave made of the Zeeman energy and the kinetic energy,but these errors are small compared to the height of thebarrier.) IV. ADDITIONAL EXAMPLES
Now we can construct some other, more interesting,examples. We will use the algebra of the group of vortexcharges to find molecules whose net charge is interestingin different ways, and we will use the energy index to test9
Components Net Charge c Stable?1 ( A, ∗ ( A,
0) (2 π, c > P − , π ) ∗ ( Q − , π ) ∗ ( R − , π ) (0 , π ) c < Q, π ) ∗ ( A,
0) ( − π , π ) c > π, , Any value MetastableTABLE I: Examples of vortex molecules. The tetrahedral charges of the components of the molecules and the net alignedcharge are given. The condition on c determines how the tetrahedra are oriented far from the vortex, due to the magnetic field.The final column indicates whether the vortex molecule is expected to have the absolute minimum energy of all vortices witha given net aligned charge. The second molecule might actually not be bound–see the text. whether they are stable.The parameter c will be less than 4 for some of theseexamples. If c < z -axis is an order2 axis. The aligned topologies have the forms ( πn, πm ) ,as described in Section I B. Of these, the only absolutelystable topologies are ± ( π, , ± (0 , π ) (and possibly( ± π, ± π ) ). A. A Doubly Quantized Pure Phase Vortex
First let us find a vortex molecule whose phase windsby 4 π . In single component condensates, such vorticesare usually unstable; one has been observed to break up,maybe into an entwined pair of 2 π vortices[30]. If phaseand spin textures were completely independent of one an-other, doubly quantized vortices would not be any morestable in the cyclic condensates; but fractional circula-tions are “bound” to certain spin textures (see Sec. I B). If we assume the vortex ( R, − π ) is stable (at the end ofSec. III A we could not decide), then a doubly-quantizedvortex can occur in a cyclic condensate when c >
4. Itconsists of the three parts( P − , π Q − , π R − , π . (78)The phase changes by 4 π while the orientation of thetetrahedron does not change at infinity as we can checkusing the coordinate system from Fig. 2b. The threegroup elements are P − = 12 (1 + i ( σ x − σ y − σ z )) Q − = 12 (1 + i ( σ x + σ y + σ z )) R − = 12 (1 + i ( σ z − σ x − σ y ))and their product is the identity.Let us discuss the conditions for binding. We havenot checked Condition 1; it is not easy to check be-cause ( P − , π ) has the same charge and energy indexas ( R, π ) ∗ ( Q, π ); an accurate solution for the spintexture around this pair of vortices is needed. Besides,( P − , π ) might be stable for some ranges of c values,but not others. Let us therefore hope that condition 1 is satisfied. Condition 3 is clear. To check condition 2,let us first consider whether one of the pairs of vorticesin the trio can coalesce. Using conservation of topolog-ical charge helps to avoid enumerating all the ways thevortices can braid around each other. If the first twovortices have coalesced into a vortex ( X, π ) (after somepermutation) and the third vortex, by winding aroundthe other two vortices as they collapsed, has changed to( Y − , π ), then ( XY − , π ) = ( id, π ) (79)by conservation of charge. Hence X = Y . Also, braid-ing one vortex between other vortices can only conjugate its group element. Therefore, Y , like R , is a counter-clockwise rotation through 120 ◦ . Since X = Y , the ro-tation part of the coalesced vortex ( X, π ) also is a 120 ◦ turn and thus the energy index of this coalesced vortexis 2 × (1 / + (4 / = 2, which is greater than the sumof the energy indices of the two vortices which formedit. Therefore the two vortices cannot coalesce sponta-neously. (This argument can be generalized to any trio ofvortices Γ , Γ , Γ each of which commutes with the netcharge Γ. Fusing two of the vortices gives the same result(up to conjugacy) no matter how the vortices are mixedaround first; so braiding cannot make a repulsive interac-tion between two vortices into an attractive one.) Finally,the three vortices cannot coalesce simultaneously because I E (0 , π ) > I E ( P − , π ) + I E ( Q − , π ) + I E ( R − , π ). B. A Vortex Molecule which is Stable
Returning to the original assumption, c >
4, where theground state orientation is illustrated by Fig. 2a, we canshow that the second charge in Eq. (72) does correspondto a completely stable vortex molecule. In fact, consider( Q, π A, . (80)This molecule, one of the decay products in Eq. (75), hasthe topology ( R − , π ) or (using the notation appropri-ate for the field-aligned tetrahedra outside the compositecore), ( − π , π ) . The energy of this molecule is approx-0imately πn ~ m [ I E ( R − , π RL q + I E ( Q, π L q a c + I E ( A,
0) ln L q a c ]= n π ~ m (ln Ra c −
16 ln L q a c ) . (81)where L q is the size of the composite core, given by Eq.(66).This molecule answers a question from Section III C.Are there stable vortices with charge ( − π , π ) ? Thetwo vortices ( − π , − π ) ∗ ( − π , π ) , have the samenet topology as a vortex of charge ( − π , π ) , and theyhave the same net energy index. Now we can check thatthe composite vortex ( Q, π )( A,
0) is a stable realizationfor the charge ( − π , π ) . Its energy is lower by a finiteamount than the energy πn π ~ m ( + ) ln Ra c of the pairof vortices. This finite binding energy is πn ~ m ln L q a .To take another point of view, the minimum-energyspin texture with the topology ( − π , π ) imposed faraway has an asymmetric structure: it has two “singular-ities” with topologies ( Q, π ) and ( A,
0) at a distance oforder L q . By contrast, when the topology imposed at aboundary corresponds to unstable vortices, the groundstate has singularities whose spacing is on the order ofthe size of the system R . E.g., in a scalar one might tryto impose ψ ( R, φ ) = √ n e iφ . The spacing of the vor-tices in the energy minimizing wave function grows with R , reflecting the fact that these vortices would repel eachother to infinity in an infinite condensate. C. A “Bound State” of No Vortices
The final example shows that point vortices are notnecessary to hold a core together–there is a “composite”vortices without any components! In other words we canconstruct a vortex for which the order parameter staysin M . There is still a “composite core” where the tetra-hedra leave M q and are no longer aligned with the fieldaxis. The trick is that the amount of rotation in a tex-ture around a vortex is defined only modulo 4 π , in theabsence of a magnetic field (because of the SU chargeclassification). When B is turned on, the spin part ofthe order parameter space M q has the same topologyas a circle, so each additional winding by 2 π changes thetopological charge. (A texture which rotates by 4 π aboutthe field axis can relax only by using axes perpendicularto the magnetic field.) Thus a 4 π -rotation-vortex is sta-ble in a magnetic field, but since it has zero tetrahedral charge, it does not have to have point vortices inside of it.(Another way to say this: Eq. (54) does not uniquely de-termine the aligned topology, because R = id . Hence a(4 π, -vortex can be made from 6 R vortices (i.e., some( R, π )’s and ( R, − π )’s) or out of no vortices at all!)Such vortices occur for both the c > c < π rotation-vortex can relaxin the absence of a magnetic field: ψ ( φ ; t ) = e − iφ ( F x sin πt + F z cos πt ) e − iF z φ √ n χ , (82)where χ is an arbitrary cyclic spinor. At each momentof time t , the expression describes an r -independent tex-ture as a function of φ . When t = 0, there is a vortexwhich is a full rotation through 720 ◦ . By the time t = 1,this vortex has completely dissipated. When q = 0 a 4 π vortex cannot relax in this way because the tetrahedra ro-tate away from the orientation preferred by the magneticfield before returning to the preferred orientation at theend. But Eq. (82) has a reincarnation as the descriptionof a (4 π, , vortex. We replace the time coordinate bya function of the radius to give a spin texture that windsthrough 4 π at infinity but does not have any singularitiesat 0: ψ = ψ ( φ ; 11 + ( rL ) ) . (83)If χ = χ or χ , then this wave function, at large r ’s,has the winding number (4 π, , . At small r ’s, thewave function is φ -independent, giving a continuous and“coreless” wave-function. (The exact solution not onlyhas a more complicated r -dependence, but also a less-symmetrical φ -dependence.) The region r . L is thecomposite core of this vortex in the sense that ψ ∈ M rather than M q . The optimal size L of this region isagain L q , as balancing the kinetic and Zeeman energiesshows.This vortex cannot disappear because the classificationof vortices at nonzero q implies that α = 4 π is conserved.Furthermore, though it does not satisfy the absolute sta-bility criterion, since two (2 π, , ’s have a smaller en-ergy, it is obviously metastable–there are no vortices inthe core to break apart! The vortex can only break upif thermal energy causes a vortex-antivortex pair to nu-cleate in the core. Suppose a pair involving rotationsthrough 2 π in opposite directions appears. These vorticesinitially attract each other but if the thermal fluctuationspull them to opposite sides of the core the nonlinear cou-pling with the background field switches this force fromattractive to repulsive, and the vortices can separate therest of the way by themselves. V. CREATING AND OBSERVING VORTEXMOLECULES
Let us discuss the conditions under which the Zeeman-effect bound states might be observed and the methodsone can use for observing them. First of all, we mustassume that q ≪ βn in order to justify neglecting q near the tetrahedral vortices and to justify the pertur-bation theory of Sec. II A. This is not just a technicalassumption: above a certain magnetic field the compo-nent vortices probably merge. To estimate the maximum1magnetic field note that q is related to the hyperfine split-ting A HF via | q | = µ B B A HF , (84)for rubidium and sodium atoms[2], and similar relationshold for other atoms. Also note that the spin independentinteraction is α = 4 π ~ ma (85)where a ∼
50 ˚A is an average of the scattering lengthscorresponding to different net spins and that the spindependent interaction is β = 4 π ~ ∆ am (86)where ∆ a ∼ differences be-tween the scattering lengths[9]. The condition for ouranalysis to be applicable, q << n β , therefore implies B << B
Max ∼ µ B r A HF ~ n ∆ am , (87)about . n = 5 × / cc.In order to observe vortex bound states, one mightstart with a condensate prepared with a spin order otherthan the ground state and then watch it evolve as in Ref.[16]. Thermal (and less importantly quantum) noise willproduce perturbations that grow exponentially, produc-ing complicated patterns. If the magnetic field is smallenough, vortex bound states might be found after sometime. To test whether these vortex bound states behavein the way we have been describing, one would have toidentify the topological charges of the vortices. One couldthen check that vortex sets whose net charge is compat-ible with the magnetic field have a size on the order ofthe theoretical value, L q . One may have to use statisti-cal correlations if too many vortices stay around. (Onecould also take a more deliberate approach, choosing vor-tex types and imprinting them as in [47] or [28]. One canthen observe the subsequent evolution of the vortices tosee whether they bind together.)In fact, identifying the vortices that appear in a spinorcondensate is difficult; vortex cores in a spinor conden-sate are not empty like the vortices in an ordinary con-densate; they have nearly the same density as the rest ofthe condensate[48]. One thus has to measure somethingabout the spins to observe the vortices. Two possibili-ties have already been invented. One can either measurethe magnetization field as in [16] or use Stern-Gerlachseparation to measure the density of the different spinspecies.Measuring the magnetization as a function of positionis less informative for a nonmagnetic phase like the cyclicphase than for the ferromagnetic phase studied in Ref. [16]. The magnetization outside the core of a vortex,where the spinor state is approximately a rotation of theunmagnetized cyclic state will be close to zero (see Eq.(45)), but inside the core, where the order parameterleaves the ground-state space M , the magnetization canbe nonzero. Measuring the magnetic moment in the coreof a vortex helps to determine the topological charge ofthe vortex. (The magnetization will not provide any di-rect evidence of the rotating orientation of the tetrahe-dral order parameter, though.) Any vortex one mighthave to identify involves a rotation about an arbitraryaxis ˆ n ′ as in Eq. (18), or Eq. (19) which is more conve-nient for understanding what a vortex will look like. Thelatter description starts with a vortex whose rotation axisis special–say it is parallel to ˆ z , and applies some overallrotation to it.For example a vortex of type ( R, π ) can be obtainedfrom a vortex whose axis is ˆ n = − ˆ z . Eq. (19) impliesthat the vortex is described by ψ ( r, φ ) = D ( R ) e i (1+ F z ) φ √ n (cid:18) f ( r ) r , , , g ( r ) r , (cid:19) T = D ( R ) √ n (cid:18) f ( r ) e iφ r , , , g ( r ) r , (cid:19) T , (88)where f ( r ) and g ( r ) are appropriate functions approach-ing 1 at infinity and R is a rotation that moves ˆ n to ˆ n ′ .(The phase, ξ , does not matter.) If ˆ n ′ = ˆ n = − ˆ z , then R is the identity, so m x ( r ) = m y ( r ) = 0 and m z ( r, φ ) = ψ ( r, φ ) † F z ψ ( r, φ )= 23 [ f ( r ) − g ( r ) ] n . (89)The magnetization is parallel to the symmetry axis and isgiven by m = n [ g ( r ) − f ( r ) ] ˆ n . Applying an arbitraryreorientation R changes the magnetization axis and thesymmetry axis in the same way, so the general result is m = 23 n [ g ( r ) − f ( r ) ] ˆ n ′ . (90)Far from the core, the magnetization vanishes. Inside thecore, the magnetization can be found by noting that thetop component of the vortex Eq. (88) must vanish at r = 0 in order to be continuous: f (0) = 0 . (91)Since α is much larger than β and γ , the density of atomswill be almost uniform across the whole vortex and hence f ( r ) + g ( r ) ≈
1. Eq. (91) therefore implies g (0) ≈ r . (92)Hence the magnetization, Eq. (90), is approximately n ˆ n ′ in the core; the atoms have a single unit of hypefinespin in the direction of the vector from the center to the2 D e n s it y PSfrag replacements .25.500 2 πn n − n n n − φ FIG. 9: Densities in the five spinor components around anorder 3 vortex, with a randomly oriented local axis ˆ n ′ . Onthe left is the pattern one might observe experimentally; thelighter regions correspond to regions with fewer atoms. Onthe right are plotted the percentage of atoms for each value of F z at some fixed distance from the vortex core. The phasesand amplitudes of these oscillations should help to determinethe direction of the local axis. fixed vertex of the rotating tetrahedra near vortex thecore[57]. The inverse vortex, ( R − , − π ), has the samemagnetization (it does not change sign). On the otherhand, similar arguments show that ( R, − π ) will have amagnetization approximately equal to − n ˆ n at the corecenter because it is the m = − A, can observe the composite vortex described in Sec-tion IV A.) Another deficiency is that the cores are onlyabout 1 µ m across, so the vortices might be hard to ob-serve directly by this method. However, one could firstallow the condensate to expand in the transverse direc-tion so that the atomic interactions decrease. The vortexcores would expand; as in experimental observations ofvortices in single-component condensates, the depletedregion in f or g (whichever corresponds to the compo-nent of the transformed spinor with the phase winding)would fly apart and the magnetized core would becomemuch larger. A magnetized ring would form at the edgeof the core where the atoms of one magnetization accu-mulate more than the atoms of the other.The Stern-Gerlach method gives more informationabout the vortices. Though the density is depleted at thecenter, the field around a vortex in a single-componentcondensate is not observable, unless one reconstructs thephase variation of the condensate, perhaps using thetechnique described in Ref. [49]. But in a spinor con-densate, the spin vortices produce observable patternsin the condensates’ Stern-Gerlach images. These imagescapture separately the density of atoms in each of the five components of the spinor as functions of position. Inthese density profiles each vortex (aside from pure phasevortices with g = id ) will be ornamented by radiatingdensity ripples as illustrated in Fig. 9. For example, ac-cording to Eq. (88), the density of atoms with F z = m is given by n ( r, φ, m ) n = | D m ( R ) r f ( r ) e iφ + D m, − ( R ) r g ( r ) | = a m + b m cos( φ − φ m ) . (93)where a m , b m are constants outside the vortex cores, since f ( r ) and g ( r ) approach 1. While a vortex in a conden-sate of a single type of atom does not show any den-sity modulation (unless the condensate interferes with asecond condensate, see e.g. Ref. [50]), angular densityripples do result for a spinor vortex as a result of the in-terference between the f and g components of the spinorproduced by of the unitary transformation changing thequantization axis from the vortex’s rotation axis ˆ n ′ tothe magnetic field direction. If the ˆ n ′ axis happens toline up exactly with the axis of the Stern-Gerlach field,then there are no radial “interference fringes,” but onlythe axes of point vortices with aligned charges will tendto line up with B . This is illustrated by the qualitativewave function in Section III. (See Eqs. (61), (62).)Both the order three and order two vortices will bevisible based on the images of the five spin components.One can determine the types of the vortices and theiraxes ˆ n ′ (which are encoded in D ( R )) from the averagemagnitude of the densities a m together with the ampli-tudes b m and offsets φ m of the density modulations. (Anorder 2 vortex will have cos 2 φ and sin 2 φ Fourier modesin addition to the terms given in Eq. (93).) A possi-ble difficulty with this method arises because, once thefive spin components are separated in space, the densityoscillations in each of them are no longer stable. Theensuing dynamics in the clouds could mix the atoms up.Distinguishing among vortices with the same rotationbut different phase winding numbers θ is not possiblewith this method without resolving the cores. For exam-ple, the vortices ( R, π ) and ( R, − π ) have the same den-sity patterns, since they differ only by an overall phase e iφ .One would also hope to check some predictions aboutthe size and charges of the bound states. One can se-lect clusters of vortices in an image of the condensate (ifthere are not too many vortices) and use the methodsjust discussed to identify the vortex charges and checkthat each cluster satisfies Eq. (54). Additionally, a signthat the vortex clusters are actually bound states is thatthe bound state size depends in the right way on the mag-netic field. Now atoms whose ground state is cyclic maybe difficult to find ( Rb is likely to be polar[39], thoughit may be possible to adjust the interaction parametersby applying light fields.). The general considerations ofthis article also apply to spin 3 condensates (see Ref.[51]), as well as to spin 1 condensates and pseudospin qn . Hence L q ∼ ~ √ mq for phases with V eff ∝ q. (94)In contrast, for the cyclic phase, Eq. (66) can be ex-pressed in terms of the scattering lengths as L q ∼ p n ∆ a ~ mq for cyclic phase (95)Since q = µ B B A HF , the size of the molecules in the cyclicphase is proportional to B and the size of molecules inphases with V eff ∝ q is proportional to B .The size of the condensate must be large enough holdan entire vortex molecule. Substituting B Max from Eq.(87) into Eqs. (94) and (95) one finds that L q ( B Max ) ∼ √ ∆ an ∼ a c (for either phase). This size is the magnetichealing length of the condensates (and the size of a vortexcore) and is on the order of 1 µ m. The condensate shouldbe narrow in one direction (so that the behavior of theorder parameter is two-dimensional) but at least severaltimes wider than the magnetic healing length in the othertwo directions; in order to measure the field dependenceof the molecule sizes, one should be able to decrease themagnetic field by some factor below B Max without themolecules leaving the condensate.As a side-comment, vortex molecules probably undergotransitions at fields close to B Max (see Fig. 10) since thecomponent vortices overlap at larger fields. Absolutelystable vortex molecules, like example 3 in Table I, willbe compressed so that the cores coincide and the vortexbecomes rotationally symmetric at a finite field. Once thecomponents’ cores overlap a little bit, being slightly offsetmight not lead to significant savings in kinetic energy.On the other hand, when the vortices in a metastable molecule are squeezed together, they form an unstabletetrahedral vortex. Metastability occurs only when the“Coulomb” force keeps the component vortices apart.
VI. CONCLUSION
We have shown that vortex molecules can be under-stood reasonably well based only on simple group the-ory and rough energy estimates. Some of these vor-tex molecules are actually metastable, and we can studytheir possible break-up “channels,” reminiscent of someof the decay processes in nuclear physics. (In practice,the molecules will probably escape through the surfaceof the condensate before any kind of “ultracold fusion”can happen!)
L 3 14 B
FIG. 10: The evolution of vortex molecules as the magneticfield is increased. The three curves illustrate how the sizes ofthe molecules from Table I, for c >
4, might change as thestrength of the magnetic field is increased. The sizes decreaseas B . At a certain magnetic field, an absolutely stable vortexwill become rotationally symmetric and the component coreswill coincide (molecule 3). Metastable vortices will becomeunstable when a certain magnetic field is reached, indicatedby the x’s terminating the curves corresponding to molecules1 and 4. More accurate calculations of the vortex fields and en-ergies could address other interesting questions. As iswell-known, unlike the Coulomb interaction between vor-tices in an ordinary scalar superfluid, the interaction en-ergy cannot be written as a sum of two-vortex interactionterms, as indicated by our estimate in terms of the energyindex. A more accurate understanding of the kinetic en-ergy landscape might show that the vortices in a moleculecan have several spatial arrangements in the core. An-other problem that requires more detailed calculations ofvortex fields is determining whether a vortex with charge( R, − π ) is stable: the energy index estimate shows thatit can break up into vortices with only a finite change inenergy, but whether the energy increases or decreases isnot clear yet. Acknowledgments
We wish to thank R. Barnett, M.Greiner, H. Y. Kee, J. Moore, K. Sengstock, D. Stamper-Kurn, and W. P. Wong for useful conversations. Wealso acknowledge financial support from CUA, DARPA,MURI, AFOSR and NSF grant 0705472.
APPENDIX A: APPENDIX: FINDING THEEFFECTIVE ACTION
Eq. (39) is not as difficult to minimize as it appears,because of the special symmetry of χ . We must substi-tute ˜ ψ = √ n χ + δ ˜ ψ , where δ ˜ ψ = dχ + aF x χ + bF y χ + cF z χ + ( e + if ) χ t (A1)into the energy, Eq. (39). The perturbation δ ˜ ψ is thedeformation of the tetrahedron, measured relative to its4body axes. Let us figure out how many powers of thecoefficients a, b, . . . to keep at each stage of calculating V . It helps to complete the square in Eq. (39) to get V tot ( ψ ) = 12 α ( ˜ ψ † ˜ ψ − n ) + 12 β ( ˜ ψ † F ˜ ψ ) + 12 γ | ˜ ψ † t ˜ ψ | − αn − q X i,j =1 cos α i cos α j ˜ ψ † F i F j ˜ ψ, (A2)where we note that the chemical potential for the cyclicstate is µ = αn and define γ = cβ . We also use F , , to stand for F x,y,z . We need to find the minimum of thisenergy only to quadratic order in q . At the end we willfind that a, b, c, . . . are each linear in q . Since each of thesquared quantities in V vanishes when a, b, c, · · · = 0, justthe linear contributions from a, b, c, . . . give the potentialto quadratic order in q . The quadratic Zeeman term,since it is multiplied by q , also is not needed beyondlinear order in a, b, c, d, e, f .Next find which matrix elements of χ need to be cal-culated to evaluate all these contributions to the en-ergy. Only the cross terms between the unperturbedpart √ n χ and the perturbation give linear functionsof a, b, c, d, e, f . For example, one cross-term containedin the quadratic Zeeman contribution is˜ ψ † F x F y ˜ ψ ≈ n χ † F x F y χ + 2 √ n ℜ χ † F x F y [( aF x + bF y + cF z + d ) χ + ( e + if ) χ t ] . (A3)Expanding this gives a sum of matrix elements suchas χ † F x F y χ and χ † t F x F y F z χ . We need only thematrix elements of products of at most three F ’s.Many of these (e.g., χ † F i F j F k χ when i, j, k are notall different, and χ † t F i F j χ when i and j are differ-ent) are equal to zero because of the 180 ◦ symme-tries of χ around the coordinate axes. The numer-ical values of the few remaining ones can be workedout quickly. Using these matrix elements to calcu-late all the terms in Eq. (A2) produces an expression V tot ( a, b, c, d, e, f, cos α , cos α , cos α ). Along the way,one notices that each of the variables a, b, c, . . . con-tributes to only one term in the q = 0 potential (thefirst line of Eq. (A2)). The variables a, b, c determinethe magnetization, d determines the density perturbationand e and f determine the singlet-amplitude θ . E.g., n = n + 2 √ n dM x = 4 √ n a (A4) ℜ [ θ ] = 2 √ n e. Finally, minimize the potential. It can be written asa sum of independent quadratic functions of a, b, c, d, e, and f : V = ( − αn + 2 qn ) + 2 αn d − √ n qd + 8 βn ( a + b + c ) + 4 √ n q ( a cos α cos α + b cos α cos α + c cos α cos α )+ 2 γn ( e + f )+ 2 q √ n [ e (cos α + cos α − α )+ √ f (cos α − cos α )] . (A5)(Note that the second term, 2 qn , is the first-order con-tribution of the Zeeman energy, which is independent oforientation.)Minimizing each quadratic (which gives a = − √ q β √ n cos α cos α , . . . ) and combining the results to-gether with the help of Eq. (37) gives V eff = − αn + 2 qn − q α + q γ − q β + ( 3 q β − q γ )(cos α + cos α + cos α );(A6)all the constant terms can be dropped to give Eq. (4).Note that the magnetization varies with the orienta-tion of the tetrahedron (as can be checked by substitut-ing the optimal values for a, b, c into the magnetization,Eq. (A4)). In particular, the c > α , cos α , cos α = ± √ has a small magnetization, m = ∓ qβ √ (1 , , ψ , it is the magnetization relative to the body axes tetra-hedron. Comparing this to the magnetic field relativeto the body axes, B (cos α , cos α , cos α ) = B √ (1 , , a, b, c, d, e, and f ; then it is eas-ier to understand vortex textures by concentrating onthe rephasing and rotation angles as a function of po-sition. The wavefunction in Eq. (11) can be parame-terized in terms of Euler angles for the rotation, e.g., ψ = √ n e iσF z e iτF x e iρF z e iθ χ . The first angle, σ , doesnot come into the angles α i that describe the orienta-tion of the magnetic field ( B ˆ z ) relative to the tetrahe-dron. (The tetrahedron can be rotated around the z -axis without changing these angles.) One can check thatcos α = sin τ sin ρ, cos α = − sin τ cos ρ, and cos α =cos τ . Now, working out the kinetic energy and combin-ing it with the effective potential gives the “phase-and-5rotation-only” energy functional E eff = Z Z d r n ~ m [2( ∇ ρ ) + 2( ∇ τ ) + 2( ∇ σ ) +( ∇ θ ) + 4 cos τ ∇ σ · ∇ ρ ]+ ( c −
4) 3 q cβ [cos τ + sin τ (sin ρ + cos ρ )] . (A7)This can be solved (in principle) to give the texturesaround sets of vortices and the relative positions of thevortices in equilibrium. Each vortex type implies a cer-tain type of discontinuity in the four angles as the core isencircled. This expression does not seem too easy to use,but at least it shows just the two effects we have beenbalancing against one another (kinetic and anisotropyenergy). The size of a vortex molecule can be esti-mated by assuming that the two terms are comparable, n ~ mL q ∼ | c − | q cβ . If distances are rescaled by L q , we thenfind that the energy function has a form that dependsonly on the sign of c −
4. Therefore, in a molecule withthree vortices, the angles of the triangle they form willbe independent of all the parameters, including c , eventhough it is dimensionless.Eq. (A7) is derived from Eq. (32) by determininghow the tetrahedra are distorted by the quadratic Zee-man effect. But the kinetic energy also causes distortionsof the wave function from the perfect tetrahedral forms,and it seems possible that these distortions could leadto kinetic effects in the anisotropy term and anisotropyeffects in the kinetic energy term. However, at the lowestorder, treating the two terms independently seems cor-rect. A simple argument for this (neglecting the kineticenergy when finding the anisotropy potential) is that thedistortion due to the Zeeman term is linear in q whilethe distortion due to the kinetic energy is quadratic in q . To see this, think of an ordinary scalar vortex, wherethe density varies as n (1 − a c r ) far from the core[2]. Theamount of “distortion” is − a c r n . In the cyclic state,distortion (i.e., perturbations to the spinor componentsthat take it out of M ) implies changes in the magneti-zation as well as the density. But we may assume thatthese distortions are still of order a c r n . The majority ofthe “pulp” in a molecule’s core consists of points whosedistance is of order L q from the actual vortex cores (the“seeds”), so the amount of distortion can be found bysubstituting r = L q from Eq. (10). Using the relationbetween a c and β shows that the fractional distortionin the “pulp” regions is of order a c r n ∼ ( qn β ) , to becompared with the deformations of order qβn that resultfrom minimizing Eq. (A5). APPENDIX B: APPENDIX:NONCOMMUTATIVITY OF VORTEX CHARGES
To give a complete description of charge conservationwhen the charges are described by the noncommutingrotations of a tetrahedron, one needs to give a rule forhow to multiply the charges of a set of vortices togetherto get the net charge. A convention we used is to multiplythe topological charges together in order of increasing x-coordinates.It seems that this definition has an awkward conse-quence: does the net vortex charge jump suddenly whentwo of the vortices are reordered, because of the non-commutativity of the group of charges? In Fig. 11 vor-tices 1 and 2 are interchanged between frames a) and c),which suggests that the net charge changes from Γ Γ Γ to Γ Γ Γ . But this deduction is incorrect, and the netvortex charge is actually conserved.The resolution of the paradox has to do with the factthat the charge of a vortex can only be determined upto conjugacy, unless one introduces a systematic conven-tion. For example, the charge of a 120 ◦ -rotation vortexis ambiguous–the rotation could be either P , Q , R , or S ,and there is no way to distinguish between these becausethe four vertices of the tetrahedra are indistinguishable.(Abstractly speaking, the four rotations are conjugate el-ements of the group.) In order to identify the charge ofeach vortex, we have to choose a routine for labelling thevertices of the tetrahedra nearby. Here is a conventionthat is consistent with the rule for ordering the vortexcharges. Take a point O far below all the vortices in thesystem and connect it with lines to points just below thevortices (see Fig. 11a). Now identify the base tetrahe-dron at O with the standard tetrahedron in Fig. 2a, mak-ing a choice from among the twelve possible ways. Thelabelling at O can be communicated to the tetrahedronat the end-point of each line, by copying the labellingfrom O to a nearby tetrahedron on the line, and thencontinuing to copy the labelling until the end of the lineis reached. Now the charges of the vortices can be iden-tified by using the labelling of the nearby tetrahedron toassign a letter to the rotation axis.Now that we have a consistent convention for assigningvortex charges, we can show that the net charge of a setof vortices does not change when two of them are inter-changed. The trick is that the charges of the individual vortices do change in such a way that the product chargedoes not change! Between Fig. 11a and Fig. 11c, vortex2 is moved over vortex 1. Because vortex 2’s tether getstangled up with vortex 1 when vortex 1 passes below it,its charge gets redefined, as Γ − Γ Γ . The other two vor-tices’ charges do not change. The net charge, obtainedby multiplying the vortex charges from left to right, isΓ Γ (Γ − Γ Γ ) = Γ Γ Γ . Thus the net charge does not change. On the other hand,there is a sudden jump in the charge of vortex 2, but thisdoes not mean that the fields of tetrahedra are changing6
PSfrag replacements
OOO Γ Γ
11 1 22 2 33 3Γ Γ Γ Γ − Γ Γ FIG. 11: a) The convention for assigning vortex charges. For a set of vortices, tethers are drawn directly from the origin toan anchor just below each vortex. As long as the tethers are moved continuously, the correspondence between the tetrahedralstate at the anchor and the standard tetrahedral state does not change. b,c) These show what happens when vortex 1 is movedbelow vortex 2. The dashed lines in a) and the dash-dot-dash lines in c) are the tethers before and after vortex 1 moves. Part b)focuses on the tether of vortex 2 , showing how the original tether gets pushed to the side by vortex 1 and is replaced by a newtether. Continuing the labelling of the tetrahedron vertices around the original path changes the labelling of the tetrahedronjust below vortex 2. Hence the charge of vortex 2 is identified differently in c). suddenly; the vortex has just been reclassified, with avortex charge that is conjugate to the original charge.Here is an interesting consequence of the noncommut-ing charges: the force between a pair of vortices changesfrom repulsive to attractive if a third vortex wanders be-tween them. In Fig. 12 one vortex (the P one) cat-alyzes a reaction without touching the other two vor-tices involved. (Assume the phases are 0 for the two B vortices and π for the P vortex.) To figure outwhat happens, keep track of when a vortex’s connec-tion to the reference point (below the figure) is inter-rupted by another vortex. The charge of the vortexpassing underneath is not changed, and the charge ofthe vortex on top changes to keep the total charge thesame. This information is sufficient for working out allthe charges: When P passes below the B on the right,the latter vortex changes to a B ′ = P − BP , so that thenet charge is still the same, even though P has moved.(This can be used to work out the charges: the netcharge of the two vortices which have switched has tobe the same, so P B ′ = BP so B ′ = P − BP .) Next,when P passes above the B on the left, the former vor-tex changes to a BP B − vortex. Now the SU matri-ces for B and P are − iσ y and − iσ x + iσ y + iσ z (using theaxes associated with χ ). Multiplying out the chargesshows that B ′ = A − . The force between the originalpair of vortices (say the P is far away at the begin-ning and end) is n ~ πmL ( I E ( B , − I E ( B, − I E ( B, n ~ πmL ( I E ( BA − ) , − I E ( A − , − I E ( B, L is thedistance between the vortices.) The vortices repel eachother at first, but after P ’s intervention, they attract eachother, as one sees by checking that BA − = C , and that B is a 2 π rotation with energy index 2 while the other PSfrag replacements
BB B P BP B − B ′ = P − BP FIG. 12: Catalysis by conjugation; the vortex on the rightmoves between the other two vortices changing their repulsionto attraction. reacting charges are all π rotations, with index .We have been assuming that q = 0, but a similar reac-tion could also happen in the core of a composite vortexwhen q = 0; that is why one has to check all the possibleways for vortices to wander between one another beforecoalescing or before dividing into groups. [1] S. Chandrasekhar (1994). [2] C. J. Pethick and H. Smith, Bose-Einstein Condensa- tion in Dilute Gases (Cambridge University Press, Cam-bridge, 2002).[3] L. P. Pitaevskii and S. Stringari, Bose-Einstein conden-sation (Clarendon Press, Oxford, 2003).[4] J. Stenger, S. Inouye, D. Stamper-Kurn, H.-J. Miesner,A. Chikkatur, and W. Ketterle, Nature , 345 (1998).[5] T.-L. Ho, Physical Review Letters , 742 (1998).[6] T. Ohmi and K. Machida, Journal of the Physical Societyof Japan , 1822 (1998).[7] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski,A. S. De, and U. Sen, Advances in Physics , 243 (2007).[8] I. Bloch, Journal of Physics B , S629 (2005).[9] C. Ciobanu, S.-K. Yip, and T.-L. Ho, Physical ReviewLetters , 033607 (2000).[10] N. D. Mermin, Reviews of Modern Physics , 591(1979).[11] K. Kasamatsu, M. Tsubota, and M. Ueda, Physical Re-view Letters , 250406 (2004).[12] T. Isoshima and S. Yip, Journal of the Physical Societyof Japan , 074805 (2006).[13] H. Makela and K. A. Suominen, Physical Review Letters , 190408 (2007).[14] C. Wu, Modern Physics Letters B , 1707 (2006).[15] D. Controzzi and A. M. Tsvelik, Physical Review Letters , 097205 (2006).[16] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore,and D. M. S. Kurn, Nature , 7109 (2006).[17] I. Chuang, R. Durrer, N. Turok, and B. Yurke, Science , 4999 (1991).[18] V. Pietila, M. Mottonen, T. Isoshima, J. A. M. Huhta-maki, and S. M. M. Virtanen, Physical Review A ,023603 (2006).[19] M. Uhlmann, R. Sch¨utzhold, and U. R. Fischer, PhysicalReview Letters p. 120407 (2007).[20] A. Lamacraft, Physical Review Letters , 160404(2007).[21] R. W. Cherng, V. Gritsev, D. M. Stamper-Kurn, andE. Demler, Physical Review Letters , 180404 (2008).[22] A. Lamacraft, Physical Review A , 063622 (2008).[23] M. Moreno-Cardoner, J. Mur-Petit, M. Guilleumas,A. Polls, A. Sanpera, and M. Lewenstein, Physical Re-view Letters , 020404 (2007).[24] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M.Stamper-Kurn, Physical Review Letters (2008).[25] K. Gross, C. P. Search, H. Pu, W. Zhang, and P. Meystre,Physical Review A , 033603 (2002).[26] R. Barnett, S. Mukerjee, and J. E. Moore (2007),arXiv:0710.5550.[27] K. Kasamatsu, M. Tsubota, and M. Ueda, InternationalJournal of Modern Physics B , 1835 (2005).[28] A. E. Leanhardt, A. Gorlitz, A. P. Chikkatur, D. Kielpin-ski, Y. Shin, D. E. Pritchard, and W. Ketterle, PhysicalReview Letters , 190403 (2002).[29] A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard,and W. Ketterle, Physical Review Letters , 140403(2003).[30] Y. Shin, M. Saba, M. Vengalattore, T. A. Pasquini,C. Sanner, A. E. Leanhardt, M. Prentiss, D. E. Pritchard,and W. Ketterle, Physical Review Letters , 160406(2004).[31] U. A. Khawaja and H. T. C. Stoof, Physical Review Ap. 043612 (2001).[32] H. Zhai, W. Chen, Z. Xu, and L. Chang, Physical ReviewA , 043602 (2003). [33] H. Stoof, E. Vliegen, and U. A. Khawaja, Physical Re-view Letters , 120407 (2001).[34] Y. Kawaguchi, M. Nitta, and M. Ueda, Physical ReviewLetters , 180403 (2008).[35] Y. Zhang, H. M¨akel¨a, and K. Suominen, Chinese PhysicsLetters , 536 (2005), see also arxiv:cond-mat/0305489.[36] R. Barnett, A. Turner, and E. Demler, Physical ReviewLetters , 180412 (2006).[37] R. Barnett, A. Turner, and E. Demler, Physical ReviewA , 013605 (2007).[38] J. Kronjager, C. Becker, P. Navez, K. Bongs, and K. Sen-gstock, Phys. Rev. Lett. , 110404 (2006).[39] M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer,K. M. Fortier, W. Zhang, L. You, and M. S. Chapman,Physical Review Letters , 140403 (2004).[40] A. Widera, F. Gerbier, S. F¨olling, T. Gericke, O. Mandel,and I. Bloch, New Journal of Physics , 152 (2006).[41] H. Schmaljohann, M. Erhard, J. Kronj¨ager, M. Kottke,S. van Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, andK. Sengstock, Physical Review Letters p. 040402 (2004).[42] D. M. Stamper-Kurn and W. Ketterle, in Coherentatomic matter waves-Ondes de matiere coherentes (LesHouches LXXII) , edited by R. Kaiser, C. Westbrook, andF. David (Springer, Berlin, 1999).[43] D. R. Nelson,
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4, the fluid is ferromagnetic–i.e., itbecomes spontaneously magnetized either parallel or an-tiparallel to the external magnetic field with an m thatcan be calculated from the expressions in Appendix A.This magnetization spontaneously breaks a symmetrysince the quadratic Zeeman effect is invariant under F z → − F z . For zero net magnetization, domains willform that have magnetization in both directions. Evenin the presence of a small cubic Zeeman term, one canshow that there is a discontinuous phase transition (asthe thermodynamical variable conjugate to magnetiza-tion is varied) although the symmetry between ± ˆ z is notexact. [55] Using the equations, the quadratic Zeeman term distortsthe ground state √ n χ with increasing magnetic field B into ψ ( B ) = √ n ( X ( B ) , , , Y ( B ) , T with X > q , Y < q because the quadratic Zeeman field favorslarge values of | F z | . The ground state √ n χ distorts into ψ ( B ) = √ n ( X ′ ( B ) , , − iY ′ ( B ) , , X ′ ( B )) with X ′ > and Y ′ < √ . The deformed state ψ ( B ) has a nonzero m and a vanishing θ and ψ ( B ) is the other way around.Incidentally, when B becomes large enough, the groundstate will become more symmetrical, with X and X ′ = 1(compare with Ref. [26]). [56] The break-up only lowers the net energy by a finite amount, about π n ~ m ln q αβ and the fragments interactwith a short-range repulsion, ∝ ln r/r perhaps; whenthe energy index of the fragments actually decreases theforce is ∝ r . Whether ( R, − π ) is stable or not we donot know, and the answer may depend on c .[57] Hence, the direction of the magnetization, ˆ nn