Topological superfluid defects with discrete point group symmetries

ABSTRACT Discrete symmetries are spatially ubiquitous but are often hidden in internal states of systems where they can have especially profound consequences. In this work we create and

verify exotic magnetic phases of atomic spinor Bose–Einstein condensates that, despite their continuous character and intrinsic spatial isotropy, exhibit complex discrete polytope symmetries

in their topological defects. Using carefully tailored spinor rotations and microwave transitions, we engineer singular line defects whose quantization conditions, exchange statistics, and

dynamics are fundamentally determined by these underlying symmetries. We show how filling the vortex line singularities with atoms in a variety of different phases leads to core structures

that possess magnetic interfaces with rich combinations of discrete and continuous symmetries. Such defects, with their non-commutative properties, could provide unconventional realizations

of quantum information and interferometry. SIMILAR CONTENT BEING VIEWED BY OTHERS TOPOLOGICAL NODAL LINE IN SUPERFLUID 3HE AND THE ANDERSON THEOREM Article Open access 17 July 2023

DENSITY-WAVE ORDERING IN A UNITARY FERMI GAS WITH PHOTON-MEDIATED INTERACTIONS Article Open access 24 May 2023 LONG-RANGE FOCUSING OF MAGNETIC BOUND STATES IN SUPERCONDUCTING LANTHANUM

Article Open access 11 September 2020 INTRODUCTION Symmetry plays a critical role in the scientific and mathematical descriptions of the universe. Symmetries can be continuous, as in

rotations of a circular cylinder about its axis; or discrete, as in end-for-end exchanges of the cylinder about its midpoint. Discrete polytope symmetries appear in diverse and widespread

systems, including crystals, molecular bonds, and the familiar morphologies of honeycombs, snowflakes, and flower petals. They can also be hidden in the internal states of otherwise

continuous and isotropic systems, where they can have profound and unusual consequences; for example, the discrete symmetries of charge conjugation, parity, and time-reversal play important

roles in particle and condensed matter physics and serve as a touchstone for grand unified theories. Complex discrete symmetries also appear in spatially uniform condensed matter systems

and, intriguingly, in spinor superfluids. In quantum mechanics, the internal symmetries of a spin-_F_ system can be conveniently described in a geometrical representation due to Majorana,

wherein a state corresponds to a constellation of 2_F_ points on the unit sphere1,2. Each point is related to the state of an independent spin-1/2 system3, and the polytopes with vertices

established by the representative points display the discrete symmetries of the order parameter that describes the system4,5,6. The stationary, dynamically stable states that share a given

constellation are known as magnetic phases, whose richness of internal symmetries is illustrated by the examples in Fig. 1. In contrast to the ubiquity of their crystalline counterparts,

condensed matter systems exhibiting discrete polytope symmetries within their internal degrees of freedom are relatively unusual, with examples appearing in exotic contexts such as _d_-wave

superconductors7, 3P2 neutron star superfluids8, and biaxial nematic liquid crystals9. Spinor Bose–Einstein condensates (BECs)10 with spin _F_ ≥ 2 provide an exciting pristine system with

unprecedented experimental control. They are described by an order parameter whose complex broken internal symmetries lead to magnetic phases with polygonal (e.g., Fig. 1f), tetrahedral

(e.g., Fig. 1g), octahedral (_F_ ≥ 3), and icosahedral (_F_ ≥ 6) Majorana symmetries. Previous experimental studies have examined phenomena associated with spinor BECs in simple magnetic

phases that lack such symmetries, including topological defects11,12,13,14 and textures15,16,17,18,19, spontaneous pattern formation20,21, dipolar interactions22, magnons23, and condensate

fragmentation24. However, apart from early experiments on spin population dynamics25, magnetic phases with discrete polytope symmetries remain unexplored. Here, we create and verify the

discrete polytope symmetries of biaxial nematic (BN) and cyclic (C) phases of spin-2 BECs in the rotational properties of their continuous wave functions and in the singular line defects.

The BN and C magnetic phases exhibit discrete internal Majorana symmetries of a square (Fig. 1f) and a tetrahedron (Fig. 1g), respectively, which are revealed experimentally. Such phases

support exotic singular vortices that we prepare using combinations of magnetic field rotations and carefully engineered microwave transitions. The singular defect cores are filled by a

variety of different superfluid phases, resulting in rich combinations of core structures that form interfaces between different discrete and continuous symmetries. Our experimental

procedure thus establishes a fascinating setting for the exploration and manipulation of the unusual properties of the line defects26,27,28,29,30 that directly emerge from the discrete

polytope symmetries. For instance, the line defects may not commute when the vortex positions are interchanged, leaving rung vortices behind in collisions30,31 with possible ramifications

for interferometry and quantum information32. RESULTS We create line defects by applying a carefully tailored time- and spatially varying magnetic field to a spin-1 87Rb superfluid in its

ferromagnetic (FM) phase, which maximizes the spin magnitude, with the order parameter defined by spatial rotations (Fig. 1a). The changing magnetic field imprints a nonsingular vortex

texture with a tight bending of the magnetization in the vicinity of its core, triggering an instability that induces the system to decay into a pair of singly-quantized, singular vortices

in the FM phase13,33. During the decay process the vortex cores fill with superfluid in the polar (P) magnetic phase, with nematic symmetry (Fig. 1b) and vanishing condensate spin magnitude.

The magnetic spin-2 superfluid phases are subsequently introduced with a sequence of microwave pulses that promote the spin-1 atoms into the spin-2 hyperfine manifold. The result is a pair

of line defects in any of the prototypical spin-2 magnetic phases shown in Fig. 1c–g, but we focus here on converting the FM phase to either the BN or C phase that display discrete polytope

symmetries. In the same pulse sequence, the filled core in the P phase is similarly converted to a distinct _F_ = 2 magnetic phase of our choice, such as ferromagnetic-2 (FM2), with maximal

spin magnitude and symmetry related to the spatial rotations, or uniaxial nematic (UN), with vanishing condensate spin and the symmetry defined by an unoriented axis and a 2_π_ change of the

global BEC phase. (Further details of the magnetic phases are provided in the Methods and in Supplementary Fig. 1.) Our experimental creation technique accesses an entire family of line

defects in which the polytope symmetries affect the properties of the vortex cores. An example of a microwave pulse sequence that yields a pair of BN vortices with filled UN cores is shown

in Fig. 2a. In the chosen basis the BN condensate is represented by a square in the _u__v_- (horizontal) plane of the Majorana sphere (Fig. 1f), whereas the UN core is similarly represented

by a rod oriented along the _w_- (vertical) axis (Fig. 1e). More generally, changing the relative phase between the nonzero spinor components of either the BN or C phases (Fig. 1f, g)

amounts to rotating the corresponding polytope about the _w_-axis, and is achieved experimentally by selecting the appropriate phases of the microwave pulses. The formation of the vortex

core can be understood in this example of the singly-quantized BN line defect as the continuous transformation of the BN bulk phase towards the UN phase inside the vortex core (Fig. 2b). We

can parameterize the stationary spinor solution by $$\zeta=\left(\begin{array}{l}{e}^{i\phi }f(\rho )/\sqrt{2}\\ 0\\ \sqrt{1-{[f(\rho )]}^{2}}\\ 0\\ {e}^{i\phi }f(\rho

)/\sqrt{2}\end{array}\right)\,,$$ (1) where _f_(_ρ_) parameterizes the radial vortex-core profile in terms of the radial coordinate \(\rho={({x}^{2}+{y}^{2})}^{1/2}\), such that _f_(_ρ_) → 1

outside the core, and _f_(_ρ_) = 0 on the line singularity itself (_ρ_ = 0). This solution smoothly interpolates between the BN and UN phases. Hence, the total superfluid density remains

non-zero as the vortex line is filled with the UN phase, even though the BN phase becomes singular and must vanish. This is unlike a defect singularity in a scalar superfluid that has a

vanishing superfluid density. The solution given by Eq. (1) thus constitutes a stable, coherent topological interface at the vortex core33,34. Vortex cores with complex internal structures

in the absence of discrete polytope symmetries have been extensively studied in superfluid liquid 3He35 and also in atomic spinor condensates36. Additionally, topological interfaces more

generally can support non-trivial topological objects whose classification changes as they penetrate the interface. These have been extensively studied, e.g., in high-energy physics and

cosmology37,38, in superfluid liquid 3He39, and in BECs34. Further instructive examples are illustrated in Fig. 2c, d, in which the core of a C line defect is filled with either the FM2 or

the UN phase (see Supplementary Figs. 2–4 for additional experimental realizations). The Majorana representation of the C phase in these cases yields a vertex-up tetrahedron with a face

parallel to the _u__v_-plane. While the FM2 core is rotationally symmetric, the core of the resulting singly-quantized C vortex is remarkably anisotropic when the UN phase occupies the line

singularity. Numerical simulations show that the vortices in both cases are unstable against splitting into a pair of vortices carrying fractional \(\frac{1}{3}\) and \(\frac{2}{3}\)

circulation quanta, respectively. In the process, the \(\frac{2}{3}\) vortex develops a FM2 core, while the core of the \(\frac{1}{3}\) vortex exhibits the ferromagnetic-1 (FM1) phase (cf.,

Fig. 1c, d). However, the decay is sensitive to the orientation of the order parameter of the imprinted vortex. In simulations, we find that a _π_/4 spin rotation about the _v_ axis instead

causes a decay of the condensate into the BN phase, forming vortices with FM2 cores. Images of the condensate spinor components reveal the rich structure of line defects in the magnetic

phases with polytope symmetries but do not directly show the discrete internal symmetries of the phases in real space. To verify these symmetries we apply radio-frequency (rf) spin-tip

pulses that rotate the entire Majorana constellation about an axis in the _u__v_-plane1, thereby preserving the magnetic phase but changing its spinor representation. The rotation axis and

angle of rotation are established by the phase and area of the rf pulse, respectively, in the same fashion as they act on states on the Bloch sphere2. Critically, the spinor component

densities remain invariant under any 2_π_/_j_ adjustments of the rf phase along a _j_-fold symmetry axis of the polytope. Extending our previous example, we demonstrate the three-fold

discrete symmetry of the cyclic magnetic phase by rotating the vertex-up tetrahedral state of the C phase (Fig. 1g) through the tetrahedral angle \(\arccos (1/\sqrt{3})\) about the axis in

the _u__v_-plane established by the phase of the spin-tip pulse. As the rf phase is varied through \(\frac{2\pi }{3}\) radians (Fig. 3b–e), the final vortex state undergoes one corresponding

revolution that returns to the edge-up state \(\frac{1}{2}{(1,\;0,\;i\sqrt{2},\;0,\;1)}^{T}\), as shown in Fig. 3b, d. The imaginary amplitude of the _m_ = 0 component, inferred from the

measured spinor component densities after the rotation, is critical to distinguishing the cyclic magnetic phase from a mixed UN-BN phase that has the same spinor component densities but does

not share the tetrahedral Majorana symmetry. Similar measurements reveal the four-fold Majorana symmetry of the BN phase under _π_/2 rotations (Fig. 4) and the full three-fold symmetry of

the C phase under 2_π_/3 rotations (Supplementary Fig. 5). DISCUSSION Our creation of line defects in magnetic phases with demonstrated polytope internal symmetries suggests a number of

future experiments. As a proof of principle, we experimentally explored the time evolution of a UN core, C vortex condensate for the first 30 ms after a tetrahedral rf rotation into the

edge-up tetrahedral state (Figs. 3b, d and 5). Over this interval, the total number of atoms decreases substantially, whereas the fraction of atoms in each spinor component changes only

slightly: the _m_ = ± 1 spinor components become more diffuse in the region of the vortex core, and the fraction of atoms in the _m_ = 0 component diminishes slightly with respect to the _m_

 = ± 2 spinor components. These results suggest that the C superfluid trends slowly towards the BN phase, consistent with previous experimental studies25. The relatively short superfluid

lifetime is a constraint that can be addressed with the development of optical trapping geometries that reduce the atomic density. Overcoming this technical hurdle will therefore establish a

path to exploring the vortex dynamics more fully, including the expected evolution towards fractionally-quantized vortices we highlighted above. Our engineering of magnetic phases with

discrete polytope symmetries and the associated line defects also opens new avenues for the further exploration of exciting phenomena that have previously only been associated with exotic

models in field theories of high-energy physics and cosmology. For instance, defects across the interfaces of polytope symmetric magnetic phases can combine non-trivially, forming

singularities that penetrate continuous, coherent topological interfaces and non-Abelian line defects appear as cosmic strings in theories of the early Universe40. Non-Abelian defects in the

magnetic phases of discrete polytope symmetries may furthermore form a non-commutative algebra26,27,28,29,30 that could potentially be harnessed for quantum information processing32.

Experiments for manipulating such defects are challenging but specific proposals for preparing pairs of non-commuting vortices already exist30. Experimental manipulation may also be assisted

by working in, e.g., strongly oblate trapping geometries where the defects become more easily controllable point vortices. METHODS SPIN-2 HAMILTONIAN We represent the macroscopic spin-2 BEC

wave function as \({{\Psi }}({{{{{{{\bf{r}}}}}}}},\;t)=\sqrt{n({{{{{{{\bf{r}}}}}}}},\;t)}{e}^{i\tau }\zeta\), such that the atom density _n_ = ∣Ψ∣2, the global phase _τ_(R, _t_), and the

spinor _ζ_(R, _t_) are governed by a mean-field Hamiltonian density10 $${{{{{{{\mathcal{H}}}}}}}}= \frac{{\hslash }^{2}}{2{M}_{{{{{{{{\rm{a}}}}}}}}}}|\nabla {{\Psi

}}{|}^{2}+{U}_{{{{{{{{\rm{trap}}}}}}}}}n+\frac{{c}_{0}}{2}{n}^{2}+\frac{{c}_{1}}{2}{n}^{2}|\langle \hat{{{{{{{{\bf{F}}}}}}}}}\rangle {|}^{2}\\

+\frac{{c}_{2}}{2}{n}^{2}|{A}_{20}{|}^{2}+pn\langle {\hat{F}}_{z}\rangle+qn\langle {\hat{F}}_{z}^{2}\rangle$$ (2) in the harmonic trapping potential

\({U}_{{{{{{{{\rm{trap}}}}}}}}}=({M}_{{{{{{{{\rm{a}}}}}}}}}{\omega }_{r}^{2}/2)({x}^{2}+{y}^{2}+2{z}^{2})\), with radial trap frequency _ω__r_. Here, _M_a denotes the atomic mass,

\({A}_{20}=\frac{1}{\sqrt{5}}(2{\zeta }_{+2}{\zeta }_{-2}-2{\zeta }_{+1}{\zeta }_{-1}+{\zeta }_{0}^{2})\) is the amplitude of spin-singlet pair formation, and _p_ and _q_ determine,

respectively, the linear and quadratic Zeeman shifts. The condensate-spin expectation value \(\langle \hat{{{{{{{{\bf{F}}}}}}}}}\rangle={\sum }_{\alpha \beta }{\zeta }_{\alpha

}^{{{{\dagger}}} }{\hat{{{{{{{{\bf{F}}}}}}}}}}_{\alpha \beta }{\zeta }_{\beta }\) is obtained from the vector \(\hat{{{{{{{{\bf{F}}}}}}}}}\) of spin-2 matrices. The nonlinearities _c_0 = 

4_π__ℏ_2(3_a_4 + 4_a_2)/7_M_a, _c_1 = 4_π__ℏ_2(_a_4 − _a_2)/7_M_a, and _c_2 = 4_π__ℏ_2(3_a_4 − 10_a_2 + 7_a_0)/7_M_a are derived from the _s_-wave scattering lengths _a__s_ for the total

spin _s_ of the interacting atom pairs, _a_0 = 87.4(10)_a_B, _a_2 = 92.4(10)_a_B, and _a_4 = 100.5(10)_a_B in units of the Bohr magneton _a_B41. For a weak quadratic Zeeman shift, the spin-2

BEC exhibits different magnetic phases that are characterized by the values of \(|\langle \hat{{{{{{{{\bf{F}}}}}}}}}\rangle|\) and ∣_A_20∣. For a ferromagnetic phase, ∣_A_20∣ = 0 and

\(|\langle \hat{{{{{{{{\bf{F}}}}}}}}}\rangle \vert=2\) (for FM2) or 1 (for FM1). The representative spinors are obtained by rotations of (1, 0, 0, 0, 0)_T_ and (0, 1, 0, 0, 0)_T_,

respectively. For the C phase, \(|\langle \hat{{{{{{{{\bf{F}}}}}}}}}\rangle \vert=\vert {A}_{20} \vert=0\) and the spinor representations are obtained by rotations of

\(\frac{1}{\sqrt{3}}{(1,\,0,\,0,\,\sqrt{2},\,0)}^{T}\). For spin-2 BECs, the polar magnetic phase, which also has a vanishing spin magnitude \(|\langle \hat{{{{{{{{\bf{F}}}}}}}}}\rangle

\vert=0\) but \(|{A}_{20} \vert=\frac{1}{\sqrt{5}}\), is found to separate into two distinct phases: UN and BN, represented by the spinors (0, 0, 1, 0, 0)_T_ and

\(\frac{1}{\sqrt{2}}{(1,\,0,\,0,\,0,\,1)}^{T}\), respectively. By defining the amplitude of spin-singlet trio formation10 \({A}_{30}=\frac{3\sqrt{6}}{2}({\zeta }_{+1}^{2}{\zeta }_{-2}+{\zeta

}_{-1}^{2}{\zeta }_{+2})+{\zeta }_{0}(-6{\zeta }_{+2}{\zeta }_{-2}-3{\zeta }_{+1}{\zeta }_{-1}+{\zeta }_{0}^{2})\) we can distinguish between the UN and BN phases that take the values

∣_A_30∣ = 1 and 0, respectively. For the C phase \(|{A}_{30} \vert=\sqrt{2}\). We solve the five coupled Gross-Pitaevskii equations obtained from Eq. (2) numerically using a split-step

method42 and experimental parameter values. Angular momentum is conserved in _s_-wave scattering, implying that on time scales where this dominates, longitudinal magnetization, _M__z_ = 

(1/_N_)∫d3_r_ _n_(R)_F__z_(R), where _N_ is the total number of atoms, is conserved. We employ an algorithm that explicitly conserves _M__z_. GRAPHICAL ILLUSTRATION OF MAGNETIC PHASE

SYMMETRIES We represent the order-parameter symmetry of the magnetic phases in terms of the complementary spherical-harmonics and Majorana representations. The stars of the Majorana

constellation1 are calculated by numerically finding the 2_F_ + 1 roots _z_ = _z__j_ of the complex polynomial, $$\mathop{\sum }\limits_{\alpha=0}^{2F}\sqrt{\left(\begin{array}{l}2F\\ \alpha

\end{array}\right)}\,{\zeta }_{F-\alpha }^{*}{z}^{\alpha }=0,$$ (3) where each root then represents a stereographic projection \({z}_{j}=\tan \left(\frac{\theta }{2}\right){e}^{i\phi }\) of

the spherical coordinates (_θ_, _ϕ_) that define the Majorana points. Rotations of the state are equivalent to rotations of the constellation on the Majorana sphere that preserve their

relative orientation. In the spherical harmonics representation we show ∣_Z_(_θ_, _ϕ_)∣2, where \(Z(\theta,\; \phi )=\mathop{\sum }\nolimits_{m=-F}^{+F}{Y}_{F,m}(\theta,\;\phi ){\zeta

}_{m}\) expands the state in terms of the spherical harmonics _Y__F_,_m_(_θ_, _ϕ_) (_F_ = 1, 2). The symmetry is completed by the color scale that is obtained from

\({{{{{{{\rm{Arg}}}}}}}}(Z)\). The spherical-harmonics representation can be obtained by representing the spin-_F_ condensate wavefunction in terms of a complex polynomial of degree _F_28.

MAGNETIC PHASES AND SINGULAR VORTEX LINES The spin-2 FM2/1 and UN phases are closely related to the spin-1 ferromagnetic (FM) and polar (P) phases, where the order parameter of the FM phase

corresponds to the group of spatial rotations, SO(3). The singular line defects in such a system, as in the FM1 phase of spin-2, can only belong to two topologically distinct equivalence

classes, representing the winding numbers zero and one13. The FM2 phase in the spin-2 system doubles the number of these classes to four. The UN phase is determined by the nematic axis

\(\hat{{{{{{{{\bf{d}}}}}}}}}\) and the global phase of the macroscopic condensate wave function _τ_. The order parameter symmetry is \({S}^{1}\times {S}^{2}/{{\mathbb{Z}}}_{2}\), where the

two-element group factorization is due to the identity of the states \(\zeta (\hat{{{{{{{{\bf{d}}}}}}}}},\;\tau )=\zeta (-\hat{{{{{{{{\bf{d}}}}}}}}},\;\tau )\). This nematic symmetry allows

for the existence of spin half-quantum vortices (HQVs). The BN and C phases exhibit polytope symmetries, as shown in Fig. 1f–g, that result in a much richer structure of singular line

defects. The topologically distinct families of line defects in such magnetic phases derive from the conjugacy classes of the fundamental homotopy group _π_1 of the corresponding

order-parameter space symmetries. Specifically, these are determined directly from the group of transformations that leave the order parameter unchanged43—here by applying a SO(3) spin

rotation and a _S_1 gauge transformation by the global condensate phase _τ_. For the case of the BN phase, the fourfold symmetry of the Majorana constellation is illustrated in Fig. 1f.

Transformations that exactly interchange the lobes in the spherical-harmonics representation and also, where necessary, take _τ_ → _τ_ + _π_, then leave the order parameter unchanged. These

transformations thus combine the dihedral-4 subgroup of SO(3) with a _π_ shift of _τ_ to form the eight-element group \({\tilde{{{{{{{{\rm{D}}}}}}}}}}_{4}\), which factorizes _S_1 × SO(3).

After lifting SO(3) to SU(2) to form a simply-connected covering group, the conjugacy classes of _π_1 are obtained30: {(_n_, 1)}, {(_n_, − 1)}, {(_n_, ± _i__σ__x_), (_n_, ± _i__σ__y_)}, 

{(_n_, ± _i__σ__z_)}, {(_n_ + 1/2, _σ_), (_n_ + 1/2, − _i__σ__z__σ_)}, {(_n_ + 1/2, − _σ_), (_n_ + 1/2, _i__σ__z__σ_)}, and {(_n_ + 1/2, ± _i__σ__x__σ_), (_n_ + 1/2, ± _i__σ__y__σ_)}, where

the Pauli matrices _σ__x_,_y_,_z_ and \(\sigma \equiv \left({{{{{{{\bf{1}}}}}}}}+i{\sigma }_{z}\right)/\sqrt{2}\) represent the SU(2) part of the fundamental homotopy group elements, and _n_

in the _S_1 part is an integer. For _n_ = 0, these represent the following topologically distinguishable vortex states: (_i_) the vortex-free state, (_i__i_) integer spin vortex,

(_i__i__i_) & (_i__v_) spin HQVs, (_v_) HQV with _π_/2 spin rotation, (_v__i_) HQV with 3_π_/2 spin rotation, and (_v__i__i_) HQV with _π_ spin rotation. For _n_ = 1, the

singly-quantized vortices in Fig. 2a, b arise from (_i_). The vortex classes in the C phase are determined through an analogous analysis27 by noting that the order parameter is left

invariant by transformations in the 12-element group \(\tilde{T}\) that combines the tetrahedral subgroup of SO(3) with elements of _S_1. This reveals the vortex classes {(_n_, 1)}, {(_n_, −

 1)}, {(_n_, ± _i__σ__α_)}, \(\{(n+1/3,\;\bar{\sigma }),\;(n+1/3,-i{\sigma }_{\alpha }\bar{\sigma })\}\), \(\{(n+1/3,\;-\bar{\sigma }),\;(n+1/3,i{\sigma }_{\alpha }\bar{\sigma })\}\),

\(\{(n+2/3,\;{\bar{\sigma }}^{2}),\;(n+2/3,-i{\sigma }_{\alpha }{\bar{\sigma }}^{2})\}\), and \(\{(n+2/3,-{\bar{\sigma }}^{2}),\;(n+2/3,\;i{\sigma }_{\alpha }{\bar{\sigma }}^{2})\}\), where

_α_ = _x_, _y_, _z_ and \(\bar{\sigma }\equiv \frac{1}{2}({{{{{{{\bf{1}}}}}}}}+i{\sigma }_{x}+i{\sigma }_{y}+i{\sigma }_{z})\). For _n_ = 0, the latter four classes represent vortices with

fractional \(\frac{1}{3}\) and \(\frac{2}{3}\) charges, while singly quantized vortices arise from the first three with _n_ = 1. EXPERIMENTAL PROCEDURES The experiment begins with

condensates of approximately 2 × 105 atoms in the \(\vert F=1,\;{m}_{F}=1\rangle\) spinor component of 87Rb. The BEC is confined in an optical trap with frequencies (_ω__r_, _ω__z_) ≈ 

2_π_(130, 170)s−1. Three pairs of Helmholtz coils generate the magnetic bias field, and a single anti-Helmholtz pair generates the magnetic quadrupole field. The condensate is initially in a

magnetic field of 17 mG along the + _z_-axis and a radial magnetic gradient of strength 4.3 G/cm. The field vanishes at a point approximately 20 μm above the condensate, and the spins

initially point along the + _z_-axis. The bias field is then reduced at a rate of 5 G/s until it reaches − 57 mG, drawing the zero point of the magnetic field through the condensate and

imprinting the nonsingular vortex as described in the main text. Subsequently, the gradient is quenched and the field is quickly reoriented to point once again along the + _z_ axis at 1.00 G

for at least 100 ms to allow the vortex to decay into two singular SO(3) vortices in the FM phase with filled P cores due to a very sharp bending of the spin texture13,33. Field drifts are

slow enough to permit adjustment to the bias fields to ensure that the magnetic field zero passes through the approximate center of the condensate. Microwave and rf pulses are applied

through a waveguide and a single coil oriented along the _y_-axis, respectively, and are initiated synchronously with the 60 Hz power line frequency to minimize the effects of ambient

time-dependent magnetic fields. A typical sequence contains more than five resonant pulses and lasts 200–400 μs, which is short compared with the typical evolution time of the condensate,

\(\sim {({c}_{0}n)}^{-1}\). Control of the magnetic fields at the milligauss level is essential, and care must be taken in the design of the pulse sequence to ensure that unwanted

transitions are not driven inadvertently because of frequency-broadening and degeneracies in the hyperfine level structure. Examples of these pulse sequences are given in Fig. 2 for BN

vortex with UN core and in Supplementary Figs. 2, 3, and 4 for BN vortex with FM2 core, C vortex with FM2 core, and for C vortex with UN core, respectively. To image the condensates, we turn

off the optical trap, whereupon the condensate falls freely and expands ballistically. A briefly applied magnetic field gradient shortly after release separates the spinor components for

simultaneous absorption imaging along the horizontal and vertical axes, which provides a pair of five atomic column density profiles for _m_ ∈ { − 2, − 1, 0, 1, 2}. The incident probe beam

is closest to resonance for _m_ = + 2 atoms to the optically excited states, and therefore the imaging efficiency of the spin-2 magnetic states decreases as the magnetic number _m_

decreases. We have calibrated this effect and normalized the atomic density of the condensate spinor components accordingly for all images and analysis in this work. In order to explicitly

demonstrate the discrete polytope point group symmetries, we control the orientation of the order parameters by inducing a spin rotation. The rotation axis and angle of rotation are

determined by the phase and area of the rf pulse. We use a pair of phase-locked direct digital synthesis (DDS) rf sources to control the frequency and phase of the applied rf and microwave

signals. One source (~0.7 MHz) directly generates the rf spin-tip pulses; the other (~30 MHz) is mixed with a carrier (6804 MHz) to generate the microwaves. Reproducible phase adjustments

are achieved by synchronizing the phase offset of both DDS sine waves at a fixed time before the pulse sequence begins. Selection of the phase for the rf spin-tip pulse achieves a rotation

of the Majorana constellation about the corresponding axis in the co-rotating _u__v_-plane defined in Fig. 3a–d. Examples of full 2_π_ rotations of the rf phase, demonstrating the discrete

polytope symmetries of the BN and C phases, are shown in Fig. 4 and Supplementary Fig. 5, respectively. DATA AVAILABILITY The relevant data generated in this study have been deposited in the

Zenodo database under accession code https://doi.org/10.5281/zenodo.6821576. CODE AVAILABILITY The code is available upon reasonable request. REFERENCES * Majorana, E. Atomi orientati in

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_Rev. Mod. Phys._ 51, 591 (1979). Article  ADS  MathSciNet  CAS  Google Scholar  Download references ACKNOWLEDGEMENTS The authors are grateful for experimental assistance from L. Reuter.

D.S.H. and T.O. thank M. Möttönen for helpful discussions. D.S.H. acknowledges financial support from the National Science Foundation (Grant No. PHY–1806318), J.R. from the UK EPSRC (Grant

Nos. EP/P026133/1, EP/S002952/1), and T.O. from the Emil Aaltonen Foundation and the Kaupallisten ja teknillisten tieteiden tukisäätiö (KAUTE) foundation through its Researchers Abroad

program. AUTHOR INFORMATION Author notes * Y. Xiao Present address: Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 48109, USA * T.

Ollikainen Present address: Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020, Innsbruck, Austria AUTHORS AND AFFILIATIONS * Department of Physics and

Astronomy, Amherst College, Amherst, MA, 01002, USA Y. Xiao, A. Blinova, T. Ollikainen & D. S. Hall * Physics, Faculty of Science, University of East Anglia, Norwich, NR4 7TJ, UK M. O.

Borgh * Department of Physics, University of Massachusetts Amherst, Amherst, MA, 01003, USA A. Blinova * QCD Labs, QTF Centre of Excellence, Department of Applied Physics, Aalto University,

P.O. Box 13500, FI–00076, Aalto, Finland T. Ollikainen * Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK J. Ruostekoski Authors * Y. Xiao View author publications You can

also search for this author inPubMed Google Scholar * M. O. Borgh View author publications You can also search for this author inPubMed Google Scholar * A. Blinova View author publications

You can also search for this author inPubMed Google Scholar * T. Ollikainen View author publications You can also search for this author inPubMed Google Scholar * J. Ruostekoski View author

publications You can also search for this author inPubMed Google Scholar * D. S. Hall View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS Y.X.,

A.B., T.O., and D.S.H. developed and conducted the experiments and analysed the data. Numerical simulations were carried out by M.O.B., and the theoretical analysis was developed by M.O.B.

and J.R. The manuscript was written by Y.X., M.O.B., J.R., and D.S.H. All authors discussed the results and commented on the manuscript. CORRESPONDING AUTHOR Correspondence to D. S. Hall.

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