Topological spinor vortex matter on spherical surface induced by non-Abelian spin-orbital-angular-momentum coupling
Jia-Ming Cheng, Ming Gong, Guang-Can Guo, Zheng-Wei Zhou, Xiang-Fa Zhou
TTopological spinor vortex matter on spherical surface induced by non-Abelianspin-orbital-angular-momentum coupling
Jia-Ming Cheng, Ming Gong, Guang-Can Guo, Zheng-Wei Zhou, and Xiang-Fa Zhou ∗ CAS Key Laboratory of Quantum Information, University of Science and Technology of China , Hefei, 230026, ChinaCAS Center For Excellence in Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, 230026, China (Dated: July 5, 2019)We provide an explicit way to implement non-Abelian spin-orbital-angular-momentum (SOAM)coupling in spinor Bose-Einstein condensates using magnetic gradient coupling. For a spherical sur-face trap addressable using high-order Hermite-Gaussian beams, we show that this system supportsvarious degenerate ground states carrying different total angular momenta J , and the degeneracycan be tuned by changing the strength of SOAM coupling. For weakly interacting spinor conden-sates with f = 1, the system supports various meta-ferromagnetic phases and meta-polar statesdescribed by quantized total mean angular momentum |(cid:104) J (cid:105)| . Polar states with Z symmetry andThomson lattices formed by defects of spin vortices are also discussed. The system can be usedto prepare various stable spin vortex states with nontrivial topology, and serve as a platform toinvestigate strong-correlated physics of neutral atoms with tunable ground-state degeneracy. I. INTRODUCTION
Spin-orbital-angular-momentum (SOAM) coupling,originally introduced due to the relativistic effect of theelectron’s spin with its orbital angular momentum, is ofubiquitousness now in varying areas of physics. For neu-tral atoms, recent investigations show that the internalatomic spin do can be coupled to its momentum degreeof freedom with the help of laser beams [1–8]. Since thefirst experimental realization of spin-momentum coupling(SMC) in the condensate of Rb atoms[3], various ex-perimental and theoretic efforts have been made alongthis direction [7–31].Most current investigations focus on the spin-momentum coupled systems. For the usual SOAM inter-action, theoretical and experimental investigations areconsidered recently only for the Abelian type interac-tion L z F z [32–41]. However, the original SOAM coupling L · F in atomic physics is non-Abelian. Such symmetricnon-Abelian feature results in various fine structures ofatomic levels with different degeneracy [42]. Meanwhile,this interaction is also closely linked with the general-ization of quantum Hall physics in 3D system [43, 44],and can be viewed as the parent system to generate al-most all relevant spin-orbital interactions discussed incurrent studies [44, 45]. However, the realization of suchnon-Abelian SOAM coupling seems extremely difficultwhich greatly constrains our abilities to explore suchnovel physics in cold atoms.On the other hand, there is also a growing interest inthe effect of the underlying geometry on various quantumorders [46–58]. For cold atoms, exotic vortex structureson a cylindrical surface have been considered recently byHo and Huang [46]. Meanwhile, spherical shell geometryinduced by hedge-hog like gradient magnetic fields has ∗ [email protected] also be proposed for spinful atoms with larger internalspin F [59]. In all these constructions, the atomic spin isfrozen along the external magnetic fields, which inhibitsthe investigation of SOAM coupled physics in curved ge-ometry. The construction of a perfect spherical surfacetrap with magnetic polarization for cold atoms also re-mains as an another experimental urgent task.In this paper, with the help of a time-dependent hedge-hog-type gradient magnetic fields (which proves to bepossible for spinful atoms)[59, 60], we show that non-Abelian SOAM coupling do can be implemented in coldatomic systems. We further show that by constructing aneffective curved surface trap using high-order Hermite-Gaussian laser beams, we can change the SOAM cou-pling strength in a wide range of parameters. Thanks tothe high symmetry of the system, the system supportsground-states with tunable degeneracy.For weakly interacting spinor condensates with f = 1[61, 62], the system support various meta-ferromagnetic(mFM) and meta-polar (mP) phases with quantized mag-nitudes of the total angular momentum (TAM) |(cid:104) J (cid:105)| = |(cid:104) L + F (cid:105)| and non-vanishing spin fluctuations. This iscompletely different from the usual vortex phases char-acterized by the quantized angular momentum L z only.In the case of vanishing spin-exchange interaction, the de-fects of spin vortices form stable lattice configurations onsphere characterised by the standard Thomson problem[59, 63]. Meanwhile, in the polar regime with strong spin-exchange interaction, the system supports stable non-trivial polar states [62, 64, 65] characterized by Z -typetopological invariant. The system can be viewed as avortex zoo of constructing stable spin vortices with novelintrinsic topology, and serve as a desired platform toexplore various non-Abelian SOAM coupled physics forboth atomic species. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l II. SCHEME OF IMPLEMENTING SOAMCOUPLING AND THE MODEL HAMILTONIAN
To illustrate the novel physics induced by such spher-ical surface trap, we consider spinor condensates suffer-ing from an isotropic non-Abelian SOAM coupling de-scribed by L · F . Such 3D SOAM coupling occurs inatomic physics due to the relativistic effect, where thespin of electrons only take values S = 1 /
2. While forneutral atoms, F can take integer and half-integer valuesfor boson and fermions, which greatly enriches the un-derlying physics. However, the implementation of suchnon-Abelian coupling is nontrivial for cold atoms as therelativstic effect is almost undetectable.To implement the isotropic SOAM coupling L · F foratoms, we introduce a Zeeman coupling term r · F involv-ing an effective hedge-hog type magnetic gradient fields.This monopole-like effective magnetic field has recentlybe proved to be possible for atoms with internal spin F [59] by employing the following Floquet engineeredquadrapole field B = B e z + B [1 − ω t )]( x e x + y e y − z e z ) (1)with a strong bias field B . The single-particle Hamilto-nian can be written as H (cid:48) = H − µ B g F B · F with H = − (cid:126) ∇ µ + V ( r ) . In current experiments, the frequency ω can be set up to10 Hz. If we choose B such that ω = Ω = µ B g F B / (cid:126) ,and assume that this frequency is much larger than allthe other energy scales, then in the rotating frame definedby U = e − i Ω tF z , the effective Hamiltonian reads H = U † H (cid:48) U − i (cid:126) U † ∂ t U = H − µ B g F B r · F , (2)where an effective monopole-like magnetic fields is in-duced.The implementation of 3D non-Abelian SOAM cou-pling is apparent now if we consider the following time-dependent Hamiltonian with a hedge-hog-type magneticZeeman coupling in the rotating frame [59, 60] H ( t ) = H + v cos(Ω t ) r · F (3)with v = − µ B g F B . Here, the driven frequency Ω (about 10 ∼ kHz) should be choosen such that ω (cid:29) Ω (cid:29) ω is satisfied. In this case, using the commutationrelation [ r · F , p · F ] = i (cid:126) ( F + L · F ), the dynamics ofthe system can be described by the following effectiveinteraction H eff = H + ˜ λ ( F + L · F ) + O (cid:18) (cid:19) , (4)where the SOAM coupling strength reads ˜ λ = v / ( µ (cid:126) Ω ). Thus, up to a constant term F , we havesucceeded in realizing the desired ( L · F )-type coupling. Although ˜ λ is always positive in the above case, we notethat negative SOAM coupling can also be implementedusing a two-step scheme with modified magnetic gradi-ent pulses. Explicit construction of these pulse sequencescan be found in Appendix A.We stress that the above expansion works only when v / Ω (cid:28) λ (cid:28)
1, which makes the non-Abelianfeature of the system hard to observe. However, as will beshown latter, the presence of a curved spherical surfacetrap in Eq.(7) can greatly enhance the effective coupling.Such spin-independent trap liberates the spin degrees offreedom, and enables the investigation of various phaseswith novel spin textures induced by non-Abelian SOAMcoupling.
III. SPHERICAL SURFACE TRAP CREATEDUSING HIGH-ORDER HERMITE-GAUSSIANBEAMS
To construct a perfect curved surface traps for spinoratoms, we propose to employ high-order Hermite-Gaussian laser beams, with which the spin degree of free-dom of the atoms can be liberated. For usual Hermite-Gaussian mode denoted as TEM mn , the electric field am-plitude reads E mn ( x, y, z ) ∝ H m ( √ xw ) H n ( √ yw ) e − x y w e ikz , (5)which is propagating along z direction with the waist ra-dius w . Here H m ( x ) is the Hermite polynomial. Thefirst few polynomials are listed as follows for later conve-nience H ( x ) = 1 , H ( x ) = 2 x, H ( x ) = 4 x − , · · · . (6) FIG. 1. Schematic plot about the implementation of thespherical surface trap using Hermite-Gaussian modes along x , y , and z -axis respectively. The inserts show the sectionalview of these modes. To obtain a spherical surface trap, we can employ com-posite Hermite-Gaussian modes along every direction.For instance, in the case of the electric dipole approx-imation, three modes such as TEM , TEM , and mod-ified TEM (see Fig. 1) along the z -axis can induce thefollowing optical potentials V ( z )00 = U e − x y w ≈ U (cid:20) − x + y ) w (cid:21) ,V ( z )11 = U x y e − x y w ≈ U x y (cid:20) − x + y ) w (cid:21) ,V ( z )20 = U x e − x y w ≈ U x (cid:20) − x + y ) w (cid:21) , where we have used the paraxial approximation such that w (cid:29) { x, y, z } . The total potential after summing overall those along the x , y and z direction reads V = (cid:88) i = x,y,z ( V ( i )00 + V ( i )11 + V ( i )20 ) ≈ U ( x + y + z )+ U ( y z + z x + x y ) − U w r + 3 U , where r = x + y + z , and other terms propor-tional to r or higher are safely neglected due to parax-ial conditions. Then if we set U = 2 U = 2 U and R = 2 U / ( U w ), the total potential field can be ap-proximated as V (cid:39) U ( r − R ) − U R + 3 U , whichindicates that the minimal value of the total potentialis obtained at r = R . Around this minimal point, totalpotential can be rewritten as (up to a constant term) V ( r ) ≈ U R ( r − R ) = 12 µω ( r − R ) (7)with µ the mass of the atom, and ω = (cid:112) U R /µ thecharacteristic frequency. We stress that to ensure theparaxial conditions, appropriate U , U and w should bechosen so that R (cid:28) w is satisfied.Eq. (7) represents a perfect spherical surface trap withtunnable radius R and trapping frequency ω inducedby laser beams. This provides an ideal platform of in-vestigating various novel physics for cold atoms subjectto such boundaryless curved geometry. Especially, forSOAM coupled condensates, the system supports exoticspinor vortex phases with intrinsic topological properties. IV. SINGLE PARTICLE SPECTRA
In the case of deep traps and low temperature,the radial motion of atoms is frozenthe and atomsare mainly confined around the spherical surface withradius R . The field operator can be assumed tobe ϕ ( r ) ψ ( θ, φ ). The radial wavefunction ϕ ( r ) reads ϕ ( r ) = (cid:0) πl T r (cid:1) − / exp (cid:2) − ( r − R ) / (2 l T ) (cid:3) , where l T = (cid:112) (cid:126) /µω (cid:28) R is the characteristic length of sphericaltrap in radial direction. After integrating out the ra-dial degree-of-freedom, we obtain a reduced dimension-less single-particle Hamiltonian in a spherical surface trapas (See Appendix B) H = L + λ L · F , (8) where λ = 2 µR ˜ λ = 2 µ B g F B R / ( (cid:126) Ω ), which thuscan be tuned in a wide range by changing the radius R ,or the ratio B respectively.The system possesses conserved quantities including L , F , J , and J z with the TAM J = L + F . Thesingle-particle eigenstates can be labeled using quantumnumbers [ l, f, j, j z ] as ψ l,fj,j z ( θ, φ ) = | j, j z (cid:105) = l (cid:88) m = − l f (cid:88) f z = − f C ( l,f ) j,j z | l, m ; f, f z (cid:105) (9)with the Clebsch-Gordan (CG) coefficients C ( l,f ) j,j z ≡(cid:104) l, m ; f, f z | j, j z (cid:105) , | l, m (cid:105) ≡ Y l,m ( θ, φ ) the usual sphericalharmonics, and | f, f z (cid:105) the internal state of spinful atoms.The corresponding single-particle energy is degeneratefor different j z and reads E = l ( l + 1) + λ j ( j + 1) − l ( l + 1) − f ( f + 1)] (10)with j = | l − f | , | l − f | + 1 , · · · , l + f . When λ > L isanti-parallel to F , and we have j = | l − f | for the groundstate. Otherwise, we have j = l + f . In both cases,the explicit values of l and j for ground states dependon the coupling λ . Therefore, the degeneracy of groundstates can be tunned in a much flexible manner. Themean values of F and L is proportial to (cid:104) J (cid:105) , and can becomputed [66] as (cid:104) F (cid:105) = 1 − α (cid:104) J (cid:105) , (cid:104) L (cid:105) = 1 + α (cid:104) J (cid:105) , (cid:104) J (cid:105) = j z e z , (11)with α = [ l ( l + 1) − f ( f + 1)] /j ( j + 1). Therefore, |(cid:104) F (cid:105)| and |(cid:104) L (cid:105)| can take fractional values for different j z .The above construction of non-Abelian SOAM cou-pling in cold atoms provides an avenue to explore var-ious novel physics with high flexibility. First, variousspin-momentum coupled subsystem in 2D can be easilyobtained by cutting the system appropriately, as shownin [44, 45]. For instance, for fixed z = z , we have a2D subsystem with Rashba-type SMC. Meanwhile, quan-tum Hall physics in 3D space can also be induced bynon-Abelian SOAM coupling [43, 44]. Second, the imple-mentation of a spherical surface trap with tunable radiusallows us to modify the strength of SOAM coupling, to-gether with the degeneracy of the ground states subspacein a much flexible way. This also enables the investiga-tion of strong-correlated physics with only a few parti-cles. Finally, for spinor condensates, this non-AbelianSOAM coupling results in various spin vortices with in-trinsic topology, as will be discussed in the following. V. PHASE DIAGRAM
For spinor condensates F = 1 with low-energy s-wavecontact scattering, the field operator ψ ( θ, φ ) containsthree components [ ψ ( θ, φ ) ψ ( θ, φ ) ψ − ( θ, φ )] T , and thereduced contact interaction in the spherical surface is H int = (cid:90) d Ω [ c : ˆ n ( θ, φ ) : + c : ˆ F ( θ, φ ) :] . (12)Here d Ω = sin θdθdφ , and :: represents the normal orderof the operator. ˆ n ( θ, φ ) = (cid:80) f z = − ψ † f z ψ f z is the totalparticle number. ˆ F = ( ˆ F + ˆ F − + ˆ F − ˆ F + ) / F z withˆ F + = ˆ F †− = √ ψ † ψ + ψ † ψ − ) and ˆ F z = ψ † ψ − ψ †− ψ − . c and c define the reduced dimensionless strengths ofdensity-density and spin-dependent interactions on thesurface trap, whose explicit forms can be written as c = g + 2 g (cid:15) N (cid:90) r | ϕ ( r ) | dr ≈ √ π a + 2 a l T N,c = g − g (cid:15) N (cid:90) r | ϕ ( r ) | dr ≈ √ π a − a l T N. Here g = 4 π (cid:126) a /µ and g = 4 π (cid:126) a /µ represent theinteractions in two-body scattering channels with totalspin F = 0, and F = 2 respectively. a and a arecorresponding s-wave scattering length. N = (cid:82) d Ωˆ n ( θ, φ )is the total number of particles.In the mean-field level, we find the phase diagramsusing both the imaginary-time-evolution and variationalmethods. Since the single-particle eigen-states are de-generate, the ground states ψ ( θ, φ ) exhibit complex spinpatterns even for condensates with weak contact interac-tion. When f = 1, the interaction energy is E int = (cid:90) d Ω [ c n ( θ, φ ) + c (cid:126) F ( θ, φ ) ] . (13)Here n ( θ, φ ) = | ψ ( θ, φ ) | is the local density and (cid:126) F ( θ, φ ) = ψ † F ψ represents the local spin-density vec-tor. Due to the symmetry, the ground state ψ ( θ, φ ) isequivalent up to a global rotation defined as R ( α, β, γ ) =exp( − iJ z α ) exp( − iJ y β ) exp( − iJ z γ ). A. Phase diagram for weak interaction c = 1 Fig.2 shows the phase diagram in the c /c − λ plainfor small quantum numbers of l and j , where the explicitground-state configurations within different regimes of λ are also provided in Tab. 1 and Appendix C. In thesecases, the ground states can be determined quantitativelyas the single-particle eigenstates have much lower degen-eracy. The phase diagram shows many novel features,which will be discussed below.First, due to the presence of interaction, the degener-acy of ψ l,fj,j z with different j z is broken even in the pres-ence of very weak interaction c = 1. For given λ and j ,the ground state carries different (cid:104) J (cid:105) (or (cid:104) L (cid:105) and (cid:104) F (cid:105) ),and supports various exotic spin patterns depending onthe ratio c /c (See Tab. I for details). Meanwhile, thephase diagram exhibits similar structures for the same j ,regardless of whether j = l + f or j = | l − f | . Therefore, -1.5 -1 -0.5 0 0.5 1-50510 FIG. 2. The mean-field ground-states of the condensates inthe c /c − λ plane. Here the phase boundaries for c = 1 areplotted with solid grey lines. “FM”,“P” and “m” are shortfor ferromagnetic, polar and meta respectively. In symbol“xx( l )”, l is the orbital-angular-momentum (OAM) quantumnumber of the condensates. “+” and “-” indicate that thespin F is parallel or anti-parallel with the OAM L . Under-lined text means that the phase supports homogeneous den-sity distributions over the surface. Other dashed lines showthe boundaries obtained for large c = 100: the red dottedline represents the boundary between polar states and mPphases; the green dot-dashed lines separate the mP phasesand other spin-vortex phases with |(cid:104) F (cid:105)| (cid:54) = 0; the boundaryfor FM phases are depicted with magenta dot-dashed lines. the whole diagram shows an approximate mirror sym-metry around the P − (1) phase with j = 0 ( l = 1) and λ ∈ (1 , l = 0 and λ ∈ ( − , c , where the magni-tude of normalized local vector | (cid:126) F ( θ, φ ) | /n ( θ, φ ) takesthe value 1 and 0 respectively. However, the presenceof SOAM coupling can result in new ground states with l (cid:54) = 0, which supports novel vortex patterns and spintextures. In addition, the abovementioned FM and polarphases appear only for strong spin-exchanging interaction | c | ∼ c . These phases also exhibit many new features,which are listed as follows:1. c /c ∼ − λ < j = l + f . Inthis case, the ground-state wavefunction reads ψ l,fj,j ( θ, φ ). Sicne j z = j , this corresponds tothe usual FM phases with maximized local vec-tor (cid:126) F ( θ, φ ) satisfying | (cid:126) F ( θ, φ ) | /n ( θ, φ ) = 1. Inaddition, the spin fluctuation defined by ∆ F ij = (cid:104)F i F j (cid:105) − (cid:104)F i (cid:105)(cid:104)F j (cid:105) with ( i, j ) ∈ ( x, y, z ) also van-ishes. These states are also denoted by FM + ( l )2. c /c ∼ − λ > j = | l − f | . Herethe ground states also reads ψ l,fj,j z = j ( θ, φ ). Since j < l , the normalized vector | (cid:126) F ( θ, φ ) | /n ( θ, φ ) < F (cid:54) = 0, and is then called as the mFM phase.3. c /c ∼ λ . In this case, the systemsupports various polar states with (cid:126) F ( θ, φ ) = 0,which exhibits novel intrinsic topological proper-ties. To show this, we write the wavefunction us-ing the real nematic order (cid:126)d as ψ ( θ, φ ) = (cid:126)d · | (cid:126)r (cid:105) = d x | x (cid:105) + d y | y (cid:105) + d z | z (cid:105) , where {| x (cid:105) , | y (cid:105) , | z (cid:105)} are theCartesian basis formed by the eigenvectors of F x,y,z with zero eigenvalues [62, 64]. For a close spheri-cal surface, the unit vector ˆ d = (cid:126)d/ | (cid:126)d | exhibits non-trivial distribution which can be described by thetopological charge W as W = 14 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dθ dφ ˆ d · ( ∂ θ ˆ d × ∂ φ ˆ d ) (cid:12)(cid:12)(cid:12)(cid:12) , (14)where we have introduced the absolute value toavoid the global ambiguity of (cid:126)d and − (cid:126)d . Fig. 3ashows that only two values 0 and 1 are allowedfor the charge W . This Z feature of W is di-rectly related to the parity of (cid:126)d ( θ, φ ), as we have (cid:126)d ( π − θ, π + φ ) → ( − l (cid:126)d ( θ, φ ) due to Y ml ( π − θ, π + φ ) → ( − l Y ml ( θ, φ ). Especially, when j = 0, thepolar phase P − (1) survives for all c /c , and therelevant vector ˆ d exhibits a stable hedge-hog likepattern with nonzero topological charge W = 1.We note that for condensates without SOAM cou-pling, such polar state is unstable towards the for-maiton of Alice ring, as shown in [62]. -1 -0.5 0 0.5012 -1 -0.5 0 0.50123 FIG. 3. (a) Z -type topological charge W of polar statesalong with λ for fixed c /c = 1, which is closely linked withthe parity of the wavefunction. See text for details; (b) and(c) shows the mean values of J along with ratio c /c fordifferent λ = − λ = − |(cid:104) J (cid:105)| can be stabilized or spoiled for larger c . Finally, across the intermediate regimes of c /c ∈ ( − , |(cid:104) J (cid:105)| (or, |(cid:104) L (cid:105)| , |(cid:104) F (cid:105)| ),as shown in Fig. 3(b-c), which represents another keyfeature of such SOAM coupled condensates. Since both |(cid:104) F (cid:105)| and the fluctuation ∆ F take nonzero values, theystill belong to the mFM phases. Beside this, the systemalso supports another mP states with |(cid:104) J (cid:105)| = |(cid:104) F (cid:105)| = 0and nonzero local spin-density vector (cid:126) F ( θ, φ ) (cid:54) = 0. Forinstance, when j = 2, λ ∈ ( − , − c /c ∈ ( − / , / ψ a ( θ, φ ) = [ √ ψ , , − ( θ, φ ) + ψ , , ( θ, φ )] / √ F ∼ I × . While for c /c > /
3, polar phase withnon-homogeneous distribution is preferred so that thespin-dependent interaction is minimized. We note thatall the abovementioned transitions are of first-order.
B. Phase diagram for strong interaction c = 10 For larger contact interaction c = 100, eigenstates ψ l,fj,j z with different j can be mixed to form new groundstates so that the density distribution of the conden-sates becomes more uniform. This leads to quantitativechanges of all the previous results.First, the boundaries for the FM and polar phasesmove leftwards on the c /c − λ plane for all phases with l (cid:54) = 0, as shown in Fig. 2 with magenta dot-dashed andred dotted lines respectively.Second, topological charge W defined in the polarregimes around c /c = 1 shift and even vanishes when λ ∈ ( − , − ψ l,fj,j z with fixed j = 2and l = 1. Additional components with different j and l should be involved.Finally, the regime of mP phases shrinks for large c and distribute mainly around the line with c = 0. Onthe other hand, the mFM regimes in the phase diagrambecomes larger. These intermediate mFM phases showcomplex patterns, and can not be simply characterisedusing quantized (cid:104) J (cid:105) (or (cid:104) L (cid:105) and (cid:104) F (cid:105) ) any more. For in-stance, new mFM phase with fixed |(cid:104) J (cid:105)| = 1 appears forintermediate c /c , as shown in Fig. 3b. While for larger | λ | = 5, the quantized feature of |(cid:104) J (cid:105)| breaks (Fig.3c),which makes the discrimination of different mFM statesto be a numerically challenging task. We leave this forfurther investigation. C. Thomson lattices formed by topological defects
In the case of neglectable spin-exchanging interactionsaround c /c ∼
0, which is fulfilled in most current ex-periments, the condensates spread almost homogeneously
TABLE I. Explicit information of different phases in figure 2 for λ < c /c . Here ”WF” is shortfor ”wavefunction”. Other symbols are defined as follows: η = (41 − √ / (cid:39) − . ψ a = [ √ ψ , , − + ψ , , ] / √ ψ c = [ ψ , , + ψ , , − ] / √
2, and ψ b = α (cid:2) ψ , , − + ψ , , (cid:3) + βψ , , with α = (cid:112) (11 c − c ) / (47 c − c ) and β = √ − α .FM(0) P(0) FM + (1) mP + (1) P + (1) FM + (2) mFM + (2) mP + (2) P + (2) c /c ( −∞ ,
0) (0 , ∞ ) ( −∞ , − /
3) ( − / , /
3) (1 / , ∞ ) ( −∞ , − /
3) ( − / , η ) ( η, /
4) (1 / , ∞ )WF ψ , , ψ , , ψ , , ψ a ψ , , ψ , , ψ , , ψ b ψ c ( |(cid:104) J (cid:105)| , |(cid:104) F (cid:105)| ) (1 ,
1) (1 ,
0) (2 ,
1) (0 ,
0) (0 ,
0) (3 ,
1) (2 , /
3) (0 ,
0) (0 , F × × × { , , } π × × { , , } π diag { , , } β α π × FIG. 4. Spin vortex defects on the spherical surface inthe meta-polar phase, where only defects with two possiblePoincar´e index Q = +1 and − over the surface so that the contact interaction is min-imized. The local vector (cid:126) F ( θ, φ ) changes its magni-tude and direction around the closed surface, with itstangential component forming different-types of defects.These defects can be characterised using the topologicalPoincar´e index Q , which only takes the value +1 or − N ± with different Q = ± N + − N − = 2 , which comes from the boundaryless feature of such sur-face trap. Around each defect center, the spin textureforms a coreless vortex [62, 65, 67]. Interestingly, for Q = −
1, we always have polar-core spin vortices with (cid:126) F ( θ, φ ) = 0 at the center. While for Q = +1, we havecoreless FM-centered vortices with the nomalized vector | (cid:126) F ( θ, φ ) | /n ( θ, φ ) ∼
1, or coreless mFM-centered vorticeswith | (cid:126) F| /n < Q = − (cid:126) F ( θ, φ ) = 0 form stable configurationscharacterised by the solution of Thomson problem for N − electrons [59, 63]. This also verifies the well-knowncharge-vortex duality for magnetic vortices in 2D system.In the case of small c = 1, the Thomson lattices are thesame when the single-particle eigenstates share the sameTAM j for given λ , as shown in Fig.5(a-c). However,for larger c = 100, the ground states can be the super- FIG. 5. Topological defects of spin vortices on sphericalsurface with charge Q = +1 and Q = − c = 1 and 100 respectively. Inall cases, defects with Q = − position of eigenstates with different j . This results innew lattice patterns for different λ , as depicted in fig-ure 5(d-h). We note that except the special case with N + = 2 and N − = 0, no such elegant Thomson patternhas been found for defects Q = +1 with nozero localvector (cid:126) F ( θ, φ ). VI. EXPERIMENTAL CONSIDERATION ANDCONCLUSION
For Rb atoms which has been widely studied in cur-rent experiments, the relevant parameters chosen in thepaper are summaried as follows: specifically, by choos-ing suitable U , U and w we can have a spherical sur-face trap with the spherical radius R = 10 µ m (cid:29) l T ≈ . µ m. The oscillating frequency of magnetic gradient B can be set to satisfy Ω = 2 π ×
50 kHz (cid:29) ω = 2 π × . λ ∈ (0 . , .
7) when B ∈ (50 , B = 6 . ω = 2 π × . (cid:29) Ω is met. Inconcrete experiments, the density ρ of Rb BECs canbe 10 cm − to 10 cm − . The total number of parti-cles N ∼ πR l T ρ in our spherical trap could reach to10 to 10 . The dimensionless interactions c can reach10 − , as required by our calculations.To summarize, we have proposed an promising routeto explore non-Ableian SOAM coupling in cold atomicsystem with the help of synthetic monopole fields. Theflexibility of the system allows us to construct an effec-tive spherical surface trap, where its ground-state de-generacy can be tuned in a wide parameter regimes.For spinor condensates with f = 1, we show that thesystem supports various exotic mFM, mP, and polarphases with nontrivial intrinsic topology. The proposedmethod works for both bosons and fermions, which thusopens up an avenue to explore various spin vortices oncurved surfaces, and may provide a new routine to inves-tigate strong-correlated physics using ultra-cold atomswith tunable ground-state degeneracy. ACKNOWLEDGMENTS
XFZ thanks Congjun Wu, Yi Li, and Shao-LiangZhang for many helpful discussions. This work wasfunded by National Natural Science Foundation of China(Grants No. 11774332, No. 11774328, No. 11574294,and No. 11474266), the major research plan of theNSFC (Grant No. 91536219),the USTC start-up funding(Grants No. KY2030000066, No. KY2030000053), theNational Plan on Key Basic Research and Development(Grant No. 2016YFA0301700), and the Strategic Pri-ority Research Program (B) of the Chinese Academy ofSciences (Grant No. XDB01030200). M.G. also thanksthe support by the National Youth Thousand Talents Program (No. KJ2030000001).
Appendix A: Spin-orbital-angular-momentumcoupling with negative sign
The realization of the SOAM coupling with negativecoefficient using gradient magnetic pulses can be dividedinto two steps.
FIG. 6. Magnetic field modulation scheme of negative SOAMcoupling.
First, using the standard magnetic pulses, such as B ∝ x e x , y e y , z e z , we can implement an intermediate 3D spin-momentum coupling as H (cid:48) eff (cid:39) H − λ (cid:48) τ µ F · P . (A1)This is possible if we consider the following sequence of magnetic pulses as U (cid:48) ( T ) = exp[ − i T (cid:126) H (cid:48) eff ] = (cid:16) e i λ (cid:48) τ (cid:126) zF z e − i T (cid:126) H e − i λ (cid:48) τ (cid:126) zF z (cid:17) (cid:16) e i λ (cid:48) τ (cid:126) yF y e − i T (cid:126) H e − i λ (cid:48) τ (cid:126) yF y (cid:17) (cid:16) e i λ (cid:48) τ (cid:126) xF x e − i T (cid:126) H e − i λ (cid:48) τ (cid:126) xF x (cid:17) ≈ exp[ − i T (cid:126) ( H − λ (cid:48) τµ F z P z )] exp[ − i T (cid:126) ( H − λ (cid:48) τµ F y P y )] exp[ − i T (cid:126) ( H − λ (cid:48) τµ F x P x )] ≈ exp[ − i T (cid:126) ( H − λ (cid:48) τ µ F · P )] , (A2)where we have assumed that the magnetic pulse is strong enough so that during the time interval τ (cid:28) T , the freeevolution of the system can be neglected.The second step is employing the hedge-hog like magnetic pulses to realize the desired SOAM coupling. Thecorresponding evolution operator reads U ( T ) = exp[ − i T (cid:126) H eff ] = e i λτ (cid:126) r · F U (cid:48) ( T ) e − i λτ (cid:126) r · F ≈ e i λτ (cid:126) r · F exp[ − i T (cid:126) ( H − λ (cid:48) τ µ F · P )] e − i λτ (cid:126) r · F ≈ exp (cid:20) − i T (cid:126) ( H − ( λ (cid:48) + 3 λ ) τ µ F · P + τ λµ ( λ (cid:48) λ L · F + F )) (cid:21) . (A3)So if we set λ (cid:48) = − λ , then the evolution operator be- comes U ( T ) = e − i T (cid:126) H eff ≈ exp (cid:20) − i T (cid:126) ( H − τ λ µ ( L · F + F )) (cid:21) , thus, we have that H eff ≈ H − τ λ µ ( L · F + F ) , (A4)which,up to a constant term F , is the desired SOAMcoupled Hamiltonian with negative coupling coefficient. Appendix B: The reduced model and eigenstates offree Hamiltonian
We consider a spin- f bosonic gas confined in a spheri-cal surface trap V ( r ) around ( R − δ, R + δ ) with δ (cid:28) R .The condensates suffer from a SOAM coupling defined by˜ λ (cid:126) L · F . For low-energy physics considered here, the ra-dial motion of bosons is frozen and its field operator canbe assumed to be ϕ ( r ) ψ ( θ, φ ). The total Hamiltoniancan be divided into two parties H tot = H + H int , (B1)in which single-particle Hamiltonian H has followingform H = (cid:90) R + δR − δ r dr (cid:90) d Ω ϕ ∗ ( r ) ψ † ( θ, φ ) ˜ H ϕ ( r ) ψ ( θ, φ ) , (B2)where d Ω = sin θdθdφ , and the Hamiltonian ˜ H containsthe SOAM coupling and reads˜ H = − (cid:126) µr ∂∂r ( r ∂∂r ) + (cid:126) L µr + ˜ λ (cid:126) L · F + V ( r ) . (B3) Here µ is atomic mass, ˜ λ stands for SOAM couplingstrength. After integrating out the radial degree-of-freedom, we obtain a reduced single-particle Hamiltonianin a spherical surface H = B (cid:126) N µR (cid:90) d Ω ψ † ( θ, φ ) H ψ ( θ, φ ) + N E , (B4)where E = (cid:82) R + δR − δ r ϕ ∗ ( r )[( − (cid:126) µ ) ∂∂r ( r ∂∂r ) + V ( r )] ϕ ( r ) dr is the energy arising from radial motion, N is the totalparticle number, and B = (cid:82) R + δR − δ r | ϕ ( r ) | dr (cid:39) ϕ ( r ). We have adopted normaliza-tion condition (cid:82) d Ω ψ † ψ = 1. Hereafter, we neglect thelast term N E by selecting a new zero energy point andset (cid:15) = B (cid:126) µR as energy unit. The free particle Hamil-tonian in the spherical surface with radius R can thenread H = L + λ L · F , with λ = 2 µR ˜ λ. (B5)Since the total angular-momentum J = L + F of thesystem is invariant, and we have [ H , J ] = 0, [ H , L ] =0, and [ H , F ] = 0. According to equality L · F = ( J − L − F ), we can derive its single-particle energy E = l ( l + 1) + λ j ( j + 1) − l ( l + 1) − f ( f + 1)] (B6)with j = | l − f | , | l − f | + 1 , · · · , l + f . So its groundstate configuration is determined by λ with the degen-eracy given by 2 j + 1. The energy of the lowest energyband for λ > E = (cid:40) [ l − λ ( f +1) − ] − [ λ ( f +1) − if l ≤ f and λ ∈ ( lf +1 , l +1) f +1 ) , [ l − λf − ] − (1 − λf ) − λf if l ≥ f and λ ∈ ( lf , l +1) f ). (B7)When λ < L is parallel to F , and we have j = l + f . The ground states energy is E = ( l + f λ + 12 ) − ( f λ + 1) λ ∈ ( − l + 1) f , − lf ) . (B8)In this way, we can figure out relations between orbital-angular-momentum quantum number l of ground states andthe corresponding spin-orbit coupling strength λ for spin f = 1 (see Fig.7).Specifically, for spinor condensates with f = 1, when λ ∈ ( − l + 1) , − l ) ( l ≥ L is parallel with F .Therefore we have j = l +1 with j z = − l − , − l, · · · , l +1.The ground-state is of 2 j + 1-fold degeneracy and can be written as ψ l, l +1 ,j z ( θ, φ ) = (cid:113) ( l + j z )( l + j z +1)(2 l +1)(2 l +2) Y l,j z − ( θ, φ ) (cid:113) l − j z +1)( l + j z +1)(2 l +1)(2 l +2) Y l,j z ( θ, φ ) (cid:113) ( l − j z )( l − j z +1)(2 l +1)(2 l +2) Y l,j z +1 ( θ, φ ) . (B9)When λ ∈ ( − , j = f = 1 and l = 0. The FIG. 7. Orbital-angular-momentum quantum number l of single-particle ground-states verses spin-orbital-angular-momentum coupling λ for spin f = 1. ground-state is 2 j + 1 = 3-fold degenerate and reads ψ , ,j z ( θ, φ ) = Y , ( θ, φ ) | f z = j z (cid:105) . (B10)When λ ∈ (1 , L of the atoms is anti-parallel with F for the ground states, so we have j = l − ψ , , ( θ, φ ) = 1 √ Y , − ( θ, φ ) − Y , ( θ, φ ) Y , ( θ, φ ) . (B11)We also address that this state supports a homogeneousdensity distribution, and the nematic vector (cid:126)d exhibitshedge-hog like pattern over the spherical surface.When λ ∈ (cid:0) l, l + 1) (cid:1) (cid:84) ( l > j = l − j z = − l + 1 , − l +2 , · · · , l −
1. The ground-state is of 2 j +1-fold degeneracyand reads ψ l, l − ,j z ( θ, φ ) = (cid:113) ( l − j z )( l − j z +1)2 l (2 l +1) Y l,j z − ( θ, φ ) − (cid:113) l − j z )( l + j z )2 l (2 l +1) Y l,j z ( θ, φ ) (cid:113) ( l + j z )( l + j z +1)2 l (2 l +1) Y l,j z +1 ( θ, φ ) . (B12) Appendix C: Ground states of the condensates for λ > When λ ∈ (1 , >
0, we have l = 1 (see Fig.7) and thetotal angular-momentum quantum number j = | l − f | =0. In this case, spin and orbital-angular-momentum arealong opposite directions. The single-particle ground-state ψ , , ( θ, φ ) (see Eq.B11) is non-degenerate, whichshould also be the ground-state for condensates withweak interaction strength. The system possesses ho-mogeneous density distribution with its spin mean-value | F | = 0 and spin-density F α ( θ, φ ) = 0 ( α = x, y, z ). So itinherently belongs to a polar state no matter whetherspin-exchange interaction is antiferromagnetic or not.Core-less vortices with vorticity n ν = − n ν = 1 appear in component f z = 1 and f z = − λ >
4, the explicit ground-states are list inTab. (II). The table shares the similar pattern as thoseshown in the main text for λ <
0, which is a reflectionof approximated mirror symmetry of the phase diagramaround the polar phase when j = 0. The explicit ex-pressions appeared in the j = 3 ( l = 4) case are list asfollows α = 14 − c c − c , β = (cid:112) − β ,η = − (cid:39) − . ,η = (880213 − √ (cid:39) − . ,η = 6741443 (cid:39) . , a = 1225 , b = 8535 . Appendix D: Spin vortices in weak interaction c = 1 Around c /c = 0, the spin-density vector (cid:126) F ( θ, φ ) = F r e r + F θ e θ + F φ e φ exhibits nontrivial patterns afterprojecting on the tangent plane of the surface. In ourcase, only spin vortices with Poincar´e index Q = ± F θ e θ + F φ e φ , as shown in Fig.4. To figure outthe distributions of the vector (cid:126) F ( θ, φ ) and the patternsformed by these spin vortex defects, we list the represen-tative spin-textures in figures 8-12. One can see that, inmost case with Q = +1, we have coreless FM-centeredvortices, or mFM-centered coreless vortices. While for Q = −
1, a polar-core spin vortex with (cid:126) F ( θ, φ ) = 0 at thecenter is favored.When λ ∈ (4 , ψ , , with max-imum mean spin F (or J , L ) and spin fluctuations.The local spin-density vector is (cid:126) F ( θ, φ ) = − [cos( θ ) e r +2 sin( θ ) e θ ] / (8 π ) as shown in Fig.10. So two mFM-centered spin vortices appear in the two poles. -1-1 -10 z xy
001 1 1
FIG. 8. Local spin-density vector (cid:126) F ( θ, φ ) in the mP + (2) phasewith SOAM coupling λ = − c = 1..Colorful symbols “+” in (a) represent spin vortices with Q =1. There are two kinds of spin vortices with N + − N − =16 −
14 = 2. TABLE II. Explicit information of different phases in figure 2 for λ > c /c . Here ”WF” is shortfor ”wavefunction”. ”D[a,b,c]” means a diagonal matrix with the diagonal elements { a, b, c } . Others are the same as those infigure 2. The explicit form of α , β , η i , a and b can be found in the context.mFM − (2) P − (2) mFM − (3) mP − (3) P − (3) mFM − (4) mFM1 − (4) mP − (4) P − (4) c c ( −∞ , ) ( , ∞ ) ( −∞ , − ) ( − , ) ( , ∞ ) ( −∞ , η ) ( η , η ) ( η , η ) ( η , ∞ )WF ψ , , ψ , , ψ , , (cid:113) ψ , , − + (cid:113) ψ , , ψ , , ψ , , ψ , , α (cid:2) ψ , , − + ψ , , (cid:3) + βψ , , (cid:113) [ ψ , , + ψ , , − ] | J | | J | = 1 | J | = 0 | J | = 2 | J | = 0 | J | = 0 | J | = 3 | J | = 2 | J | = 0 | J | = 0 | F | | F | = | F | = 0 | F | = | F | = 0 | F | = 0 | F | = | F | = | F | = 0 | F | = 0∆ F D { , , } π × { , , } π D { , , } π × { a,a,b } π D { , , } π D { , , } α β π × -1-1 -10 z xy
001 1 1
FIG. 9. Local spin-density vector (cid:126) F ( θ, φ ) in the mP + (1) phasewith SOAM coupling λ = − c = 1.Colorful symbols “+” in (a) represent spin vortices with Q =1. There are two kinds of spin vortices with N + − N − =8 − -1-1 -10 z xy
001 1 1
FIG. 10. Local spin-density vector (cid:126) F ( θ, φ ) in the mFM − (2)with SOAM coupling λ = 5 and weak interaction c = 1.Colorful symbols “+” in (a) represent spin vortices with Q =1. There are two spin vortices with Q = 1.[1] Dieter Jaksch and Peter Zoller, “Creation of effectivemagnetic fields in optical lattices: the hofstadter but-terfly for cold neutral atoms,” New Journal of Physics ,56 (2003).[2] Shi-Liang Zhu, Baigeng Wang, and L-M Duan, “Simu-lation and detection of dirac fermions with cold atoms inan optical lattice,” Physical Review Letters , 260402(2007).[3] Y.-J. Lin, K Jim´enez-Garc´ıa, and I.B. Spielman, “Spin- orbit-coupled bose-einstein condensates,” Nature ,83–86 (2011).[4] Jin-Yi Zhang, Si-Cong Ji, Zhu Chen, Long Zhang, Zhi-Dong Du, Bo Yan, Ge-Sheng Pan, Bo Zhao, You-JinDeng, Hui Zhai, et al., “Collective dipole oscillations ofa spin-orbit coupled bose-einstein condensate,” PhysicalReview Letters , 115301 (2012).[5] Wei Zhang and Wei Yi, “Topological fulde-ferrell-larkin-ovchinnikov states in spin–orbit-coupled fermi gases,” -1 -1-10 z xy
001 1 1
FIG. 11. Local spin-density (cid:126) F ( θ, φ ) in the mP − (3) withSOAM coupling λ = 7 and weak interaction c = 1. Col-orful symbols “+” in (a) represent spin vortices with Q = 1.There are two kinds of spin vortices with N + − N − = 8 − -1-1 -10 z xy
001 1 1
FIG. 12. Local spin-density (cid:126) F ( θ, φ ) in the mP − (4) withSOAM coupling λ = 9 and weak interaction c = 1. Colorfulsymbols “+” in (a) represent spin vortices with Q = 1. Thereare two kinds of spin vortices with N + − N − = 16 −
14 = 2.Nature Communications (2013).[6] Si-Cong Ji, Jin-Yi Zhang, Long Zhang, Zhi-Dong Du,Wei Zheng, You-Jin Deng, Hui Zhai, Shuai Chen, andJian-Wei Pan, “Experimental determination of the finite-temperature phase diagram of a spin-orbit coupled bosegas,” Nature Physics , 314–320 (2014).[7] Zhan Wu, Long Zhang, Wei Sun, Xiao-Tian Xu, Bao-Zong Wang, Si-Cong Ji, Youjin Deng, Shuai Chen,Xiong-Jun Liu, and Jian-Wei Pan, “Realization of two-dimensional spin-orbit coupling for bose-einstein conden-sates,” Science , 83–88 (2016).[8] Lianghui Huang, Zengming Meng, Pengjun Wang, PengPeng, Shao-Liang Zhang, Liangchao Chen, Donghao Li,Qi Zhou, and Jing Zhang, “Experimental realization oftwo-dimensional synthetic spin-orbit coupling in ultra-cold fermi gases,” Nature Physics , 540–544 (2016).[9] Victor Galitski and Ian B. Spielman, “Spin-orbit couplingin quantum gases,” Nature , 49 (2013).[10] Hui Zhai, “Degenerate quantum gases with spin-orbitcoupling: a review,” Reports on Progress in Physics ,026001 (2015).[11] Xiangfa Zhou, Yi Li, Zi Cai, and Congjun Wu, “Uncon-ventional states of bosons with the synthetic spin-orbitcoupling,” Journal of Physics B: Atomic, Molecular andOptical Physics , 134001 (2013).[12] Chunji Wang, Chao Gao, Chao-Ming Jian, and Hui Zhai,“Spin-orbit coupled spinor bose-einstein condensates,”Physical Review Letters , 160403 (2010).[13] Wu Cong-Jun, Ian Mondragon-Shem, and Zhou Xiang-Fa, “Unconventional bose-einstein condensations from spin-orbit coupling,” Chinese Physics Letters , 097102(2011).[14] Hui Hu, B. Ramachandhran, Han Pu, and Xia-Ji Liu,“Spin-orbit coupled weakly interacting bose-einstein con-densates in harmonic traps,” Physical Review Letters , 010402 (2012).[15] Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, andSandro Stringari, “Superstripes and the excitation spec-trum of a spin-orbit-coupled bose-einstein condensate,”Physical Review Letters , 235302 (2013).[16] Tomoki Ozawa and Gordon Baym, “Stability of ultracoldatomic bose condensates with rashba spin-orbit couplingagainst quantum and thermal fluctuations,” Physical Re-view Letters , 025301 (2012).[17] Valery E. Lobanov, Yaroslav V. Kartashov, andVladimir V. Konotop, “Fundamental, multipole, andhalf-vortex gap solitons in spin-orbit coupled bose-einstein condensates,” Physical Review Letters ,180403 (2014).[18] Yong-Chang Zhang, Zheng-Wei Zhou, Boris A. Mal-omed, and Han Pu, “Stable solitons in three dimensionalfree space without the ground state: Self-trapped bose-einstein condensates with spin-orbit coupling,” PhysicalReview Letters , 253902 (2015).[19] Xi-Wang Luo, Kuei Sun, and Chuanwei Zhang, “Spin-tensor–momentum-coupled bose-einstein condensates,”Physical Review Letters , 193001 (2017).[20] Jun-Ru Li, Jeongwon Lee, Wujie Huang, Sean Burchesky,Boris Shteynas, Furkan C¸ a˘grı Top, Alan O Jamison, andWolfgang Ketterle, “A stripe phase with supersolid prop-erties in spin-orbit-coupled bose-einstein condensates,”Nature , 91 (2017).[21] Wei Han, Xiao-Fei Zhang, Deng-Shan Wang, Hai-FengJiang, Wei Zhang, and Shou-Gang Zhang, “Chiral super-solid in spin-orbit-coupled bose gases with soft-core long-range interactions,” Physical review letters , 030404(2018).[22] Rukuan Wu and Zhaoxin Liang, “Beliaev damping ofa spin-orbit-coupled bose-einstein condensate,” Physicalreview letters , 180401 (2018).[23] Renyuan Liao, “Searching for supersolidity in ultracoldatomic bose condensates with rashba spin-orbit cou-pling,” Physical review letters , 140403 (2018).[24] Chunlei Qu and Sandro Stringari, “Angular momentumof a bose-einstein condensate in a synthetic rotationalfield,” Physical review letters , 183202 (2018).[25] Chunlei Qu, Zhen Zheng, Ming Gong, Yong Xu, Li Mao,Xubo Zou, Guangcan Guo, and Chuanwei Zhang, “Topo-logical superfluids with finite-momentum pairing and ma-jorana fermions,” Nature Communications (2013).[26] Hui Hu, Lei Jiang, Xia-Ji Liu, and Han Pu, “Prob-ing anisotropic superfluidity in atomic fermi gases withrashba spin-orbit coupling,” Physical Review Letters , 195304 (2011).[27] Fan Wu, Guang-Can Guo, Wei Zhang, and Wei Yi, “Un-conventional superfluid in a two-dimensional fermi gaswith anisotropic spin-orbit coupling and zeeman fields,”Physical Review Letters , 110401 (2013).[28] Subhasis Sinha, Rejish Nath, and Luis Santos, “Trappedtwo-dimensional condensates with synthetic spin-orbitcoupling,” Physical Review Letters , 270401 (2011).[29] Yi Li, Xiangfa Zhou, and Congjun Wu, “Three-dimensional quaternionic condensations, hopf invariants,and skyrmion lattices with synthetic spin-orbit cou- pling,” Physical Review A , 033628 (2016).[30] William S Cole, Shizhong Zhang, Arun Paramekanti,and Nandini Trivedi, “Bose-hubbard models with syn-thetic spin-orbit coupling: Mott insulators, spin textures,and superfluidity,” Physical Review Letters , 085302(2012).[31] Jia-Ming Cheng, Xiang-Fa Zhou, Zheng-Wei Zhou,Guang-Can Guo, and Ming Gong, “Symmetry-enrichedbose-einstein condensates in a spin-orbit-coupled bilayersystem,” Physical Review A , 013625 (2018).[32] Karl-Peter Marzlin, Weiping Zhang, and Ewan M.Wright, “Vortex coupler for atomic bose-einstein conden-sates,” Physical review letters , 4728 (1997).[33] Xiong-Jun Liu, Xin Liu, Leong Chuan Kwek, andChoo Hiap Oh, “Optically induced spin-hall effect inatoms,” Physical Review Letters , 026602 (2007).[34] Kuei Sun, Chunlei Qu, and Chuanwei Zhang, “Spin–orbital-angular-momentum coupling in bose-einstein con-densates,” Physical Review A , 063627 (2015).[35] Chunlei Qu, Kuei Sun, and Chuanwei Zhang, “Quan-tum phases of bose-einstein condensates with syntheticspin–orbital-angular-momentum coupling,” Physical Re-view A , 053630 (2015).[36] Yu-Xin Hu, Christian Miniatura, Benoˆıt Gr´emaud, et al.,“Half-skyrmion and vortex-antivortex pairs in spinorcondensates,” Physical Review A , 033615 (2015).[37] Michael DeMarco and Han Pu, “Angular spin-orbit cou-pling in cold atoms,” Physical Review A , 033630(2015).[38] H.-R. Chen, K.-Y. Lin, P.-K. Chen, N.-C. Chiu,J.-B. Wang, C.-A. Chen, P.-P. Huang, S.-K. Yip,Yuki Kawaguchi, and Y.-J. Lin, “Spin-orbital-angular-momentum coupled bose-einstein condensates,” PhysicalReview Letters , 113204 (2018).[39] Dongfang Zhang, Tianyou Gao, Peng Zou, Lingran Kong,Ruizong Li, Xing Shen, Xiao-Long Chen, Shi-Guo Peng,Mingsheng Zhan, Han Pu, and Kaijun Jiang, “Ground-state phase diagram of a spin-orbital-angular-momentumcoupled bose-einstein condensate,” Phys. Rev. Lett. ,110402 (2019).[40] P.-K. Chen, L.-R. Liu, M.-J. Tsai, N.-C. Chiu,Y. Kawaguchi, S.-K. Yip, M.-S. Chang, and Y.-J. Lin,“Rotating atomic quantum gases with light-induced az-imuthal gauge potentials and the observation of the hess-fairbank effect,” Physical Review Letters , 250401(2018).[41] Xiao-Long Chen, Shi-Guo Peng, Peng Zou, Xia-Ji Liu,and Hui Hu, “Angular stripe phase in spin-orbital-angular-momentum coupled bose condensates,” arXivpreprint arXiv:1901.02595 (2019).[42] Robert D. Cowan, The theory of atomic structure and spectra,3 (Univ of California Press, 1981).[43] Yi Li, Shou-Cheng Zhang, and Congjun Wu, “Topologi-cal insulators with su(2) landau levels,” Physical ReviewLetters , 186803 (2013).[44] Yi Li and Congjun Wu, “High-dimensional topological in-sulators with quaternionic analytic landau levels,” Phys-ical Review Letters , 216802 (2013).[45] Yi Li, Xiangfa Zhou, and Congjun Wu, “Two-andthree-dimensional topological insulators with isotropicand parity-breaking landau levels,” Physical Review B , 125122 (2012).[46] Tin-Lun Ho and Biao Huang, “Spinor condensates ona cylindrical surface in synthetic gauge fields,” Physical Review Letters , 155304 (2015).[47] Kuei Sun, Karmela Padavi´c, Frances Yang, SmithaVishveshwara, and Courtney Lannert, “Static and dy-namic properties of shell-shaped condensates,” PhysicalReview A , 013609 (2018).[48] Jian Zhang and Tin-Lun Ho, “Potential scattering on aspherical surface,” Journal of Physics B: Atomic, Molec-ular and Optical Physics , 115301 (2018).[49] A Tononi and L Salasnich, “Bose-einstein conden-sation on the surface of a sphere,” arXiv preprintarXiv:1903.08453 (2019).[50] J Batle, Armen Bagdasaryan, M Abdel-Aty, and S Ab-dalla, “Generalized thomson problem in arbitrary dimen-sions and non-euclidean geometries,” Physica A: Statisti-cal Mechanics and its Applications , 237–250 (2016).[51] Vladimir M Fomin, Roman O Rezaev, and Oliver GSchmidt, “Tunable generation of correlated vortices inopen superconductor tubes,” Nano Letters , 1282–1287(2012).[52] Oleksandr V. Pylypovskyi, Volodymyr P. Kravchuk, De-nis D. Sheka, Denys Makarov, Oliver G. Schmidt, andYuri Gaididei, “Coupling of chiralities in spin and physi-cal spaces: The m¨obius ring as a case study,” Phys. Rev.Lett. , 197204 (2015).[53] V Parente, P Lucignano, P Vitale, A Tagliacozzo, andF Guinea, “Spin connection and boundary states in atopological insulator,” Physical Review B , 075424(2011).[54] Yi Li and FDM Haldane, “Topological nodal cooperpairing in doped weyl metals,” arXiv preprintarXiv:1510.01730 (2015).[55] Ken-Ichiro Imura, Yukinori Yoshimura, Yositake Takane,and Takahiro Fukui, “Spherical topological insulator,”Physical Review B , 235119 (2012).[56] Yaacov E Kraus, Assa Auerbach, HA Fertig, andSteven H Simon, “Testing for majorana zero modes in a p x + ip y superconductor at high temperature by tunnel-ing spectroscopy,” Physical Review Letters , 267002(2008).[57] Sergej Moroz, Carlos Hoyos, and Leo Radzihovsky, “Chi-ral p ± ip superfluid on a sphere,” Physical Review B ,024521 (2016).[58] Zhe-Yu Shi and Hui Zhai, “Emergent gauge field for achiral bound state on curved surface,” arXiv preprintarXiv:1510.05815 (2015).[59] Xiang-Fa Zhou, Congjun Wu, Guang-Can Guo, RuquanWang, Han Pu, and Zheng-Wei Zhou, “Synthetic lan-dau levels and spinor vortex matter on a haldane spheri-cal surface with a magnetic monopole,” Physical ReviewLetters , 130402 (2018).[60] Nathan Goldman and Jean Dalibard, “Periodicallydriven quantum systems: effective hamiltonians and en-gineered gauge fields,” Physical Review X , 031027(2014).[61] Tin-Lun Ho, “Spinor bose condensates in optical traps,”Physical Review Letters , 742 (1998).[62] Yuki Kawaguchi and Masahito Ueda, “Spinor bose–einstein condensates,” Physics Reports , 253–381(2012).[63] Joseph John Thomson, “Xxiv. on the structure of theatom: an investigation of the stability and periods ofoscillation of a number of corpuscles arranged at equalintervals around the circumference of a circle; with ap-plication of the results to the theory of atomic structure,” The London, Edinburgh, and Dublin Philosophical Mag-azine and Journal of Science , 237–265 (1904).[64] Tilman Zibold, Vincent Corre, Camille Frapolli, AndreaInvernizzi, Jean Dalibard, and Fabrice Gerbier, “Spin-nematic order in antiferromagnetic spinor condensates,”Physical Review A , 023614 (2016).[65] Justin Lovegrove, Magnus O Borgh, and Janne Ru-ostekoski, “Stability and internal structure of vortices in spin-1 bose-einstein condensates with conserved magne-tization,” Physical Review A , 033633 (2016).[66] Morris Edgar Rose, Elementary theory of angular momentum(Courier Corporation, 1995).[67] Takeshi Mizushima, Naoko Kobayashi, and KazushigeMachida, “Coreless and singular vortex lattices in rotat-ing spinor bose-einstein condensates,” Physical ReviewA70