Synthetic half-integer magnetic monopole and single-vortex dynamics in spherical Bose-Einstein condensates

Magnetic monopoles, which are particles with isolated magnetic charges, have been a subject of speculation in theoretical physics. Although the existence of elementary magnetic monopoles remains a mystery [1, 2, 3, 4, 5, 6, 7], analogues emerge in condensed matter systems, for instance, on the interfaces of rotating superfluid mixtures [8], at the ends of spin chains in spin ices [9, 10, 11, 12] and that of skyrmion-line excitations in chiral magnets [13], or in the momentum space of quantum Hall systems [14]. Monopoles can also be mimicked by creating a thin magnetic needle [15], the Berry curvature of a varying superconducting qubit state [16], or through magneto-optical effects [17].

The existence of magnetic monopoles implies that the magnetic field is no longer sourceless, i.e., $\nabla\cdot\bm{B}\neq 0$. To retain the electromagnetic vector potential $\bm{A}(\bm{r})$ in this scenario, it is necessary to introduce a singular line in $\bm{A}(\bm{r})$, known as the Dirac string. The Dirac string shows its physical effects, when a charged particle (with electric charge $q_{e}$) moves around the string closely. The particle acquires an Aharonov-Bohm phase $e^{iq_{e}\Phi}$, where $\Phi=4\pi g_{m}$ is the magnetic flux ($g_{m}$ is the magnetic charge) passing through a surface (which does not intersect the Dirac string) bounded by the particle’s trajectory, as illustrated in Fig.1. Crucially, since the Dirac string is hypothetical and unobservable, it must yield no detectable consequences. This requires the phase factor to satisfy $q_{e}\Phi=2n\pi$ with $n\in\mathbb{Z}$ (in natural units), leading directly to the Dirac charge quantization condition: $2g_{m}q_{e}\in\mathbb{Z}$ [1, 2]. This duality explains why electric charges are quantized in the universe if a magnetic monopole exists, i.e. $q_{e}=n/2g_{m}$. Conversely, it also restricts magnetic charges to be integer multiples of $1/2q_{e}$.

In experiments of cold atom simulation with Bose-Einstein condensates (BEC), the effective magnetic field originates from the Berry curvature, which stems from the adiabatic evolution of the atomic spin state $|\chi(\bm{r})\rangle$ in space. The corresponding Berry connection, $\bm{A}(\bm{r})=\langle\chi(\bm{r})|i\nabla|\chi(\bm{r})\rangle$, serves as an effective gauge field, which introduces a geometric phase $e^{i\int\bm{A}(\bm{r})\cdot d\bm{r}}$ into the atomic spatial wavefunction, depending on the path. By analogy with the form of the Aharonov-Bohm effect, the effective charge in the geometric phase is identified to $q_{e}=1$. Synthetic magnetic monopoles can be constructed by aligning the spin of the atoms (with the quantum number $F$) with a special configuration in space so that moving atoms experience a Berry curvature corresponding to a monopole with $g_{m}=F$ [18]. From the quantization condition, the allowed magnetic charge is either an integer or a half-integer. However, magnetic monopole realized in the BEC simulations so far, contains an integer magnetic charge [18, 19, 20, 21] since the atomic spin quantum number $F$ must be an integer for bosons. Whether and how half-integer-type magnetic monopoles, especially the minimal monopole, can be realized in boson gases remains an open question.

With a magnetic monopole situated in the center of a sphere, a uniform magnetic field perpendicular to the spherical surface is created. For the noninteracting case, the single-particle wave functions on the sphere surface, the so-called monopole harmonics, have been analytically obtained by Wu and Yang [3]. The eigenstate has vortices with a total vortex charge of $2g_{m}$. Especially for the $g_{m}=1$ bosonic case, the ground state contains two vortices [3] and these vortices can be manipulated to desired position [22]. In the interacting case, for a BEC system with an integer $g_{m}$, total $2g_{m}$ vortices exist in the ground state and form stable lattice patterns, like Thomson plum pudding model [23]. So far, the vortex dynamics on the spherical BEC in the literature mainly focus on cases in the absence of magnetic monopoles, such as vortex-antivortex dynamics, BKT phase transition, and the formation of vortex clusters [24, 29, 26, 31, 32, 30, 25, 27, 28, 33, 34, 35]. Yet, it is unclear how a monopole influences the dynamics of vortex.

On the other hand, the spherical system with a monopole serves as an ideal framework for investigating fractional quantum hall effects [36] and vortex dynamics without boundary effects in a finite size [31]. It is imperative to examine the realization of the minimal monopole and the characteristics of the minimal vortex-monopole unit in the spherical system, which underpins the study of intricate vortex structures in the presence of monopoles with high magnetic charges. In this article, we explore the realization of a magnetic monopole with a half-integer magnetic charge in a cold atom system, as well as the dynamics of a single vortex (in the presence of a minimal monopole) driven by an external field. The rest of the article is organized as follows:

In Sec. II, we propose a flexible scheme to realize a magnetic monopole with a half-integer magnetic charge in the alkali- or hydrogen-atom system. The key steps are to freeze the electron’s angular momentum $(\bm{J})$ and polarize the nuclear spin (with a half-integer quantum number $I$) aligning to the direction of the atom’s position vector, leading to an effective radial magnetic field corresponding to a monopole field of $g_{m}=I$.

In Sec. III, we develop the ground state of spherical BEC in the presence of a monopole with a minimal charge $g_{m}=1/2$. For strong interaction, we take the trial wave function $\psi(\theta,\phi)\propto\tanh(b\theta)e^{i\phi}$ for the ground state with a vortex locating at the north pole (in spherical coordinates). The fitting function agrees very well with the results obtained from the numerical calculation. We generalize the solution by rotating the vortex core to an arbitrary position. We demonstrate that each atom in this vortex-monopole system possesses angular momentum of magnitude ${\hbar}/{2}$ orientating to the vortex position.

In Sec. IV, we study the vortex dynamics of spherical BEC with a monopole $g_{m}=1/2$ and an additional external electromagnetic field. Assuming a rigid vortex-monopole configuration, we demonstrate that the vortex-monopole structure performs a precession motion and develop an equation of motion to describe the trajectory of the vortex center.

In Sec. V, we employ the Gross-Pitaevskii equation (GPE) to simulate the vortex dynamics in a 3D shell-shaped system with a magnetic monopole, where a ‘bi-gauge evolution’ approach is developed to avoid singularities of the gauge fields in the numerical calculation. The numerical results are consistent with our theoretical prediction.

Our work offers a scheme for the realization of magnetic monopoles with half-integer charges, validates the point-vortex picture for the strong interaction in the presence of monopoles, and advances the study of the dynamics of the simplest vortex-monople structure.

In this section, we present the scheme of realizing monopoles with half-integer magnetic charges. Considering an atom in the presence of a magnetic field $\bm{B}$, the Hamiltonian describing the energy level splitting for the internal degree of freedom of atoms contains the hyperfine interaction and Zeeman energies,

where $A$ is a constant, $\bm{\hat{I}}$ is the nuclear spin, $\bm{\hat{J}}$ denotes the total angular momentum of electron in the atom, $g\simeq 2$ is the Landé $g$-factor, $\mu_{B}$ is the Bohr magneton, and $\mu_{I}\bm{\hat{I}}/I$ is the nuclear magnetic moment [37]. The total angular momentum of electron is $\bm{\hat{J}}=\bm{\hat{L}}+\bm{\hat{S}}$, where $\bm{\hat{L}}$ is the electron’s orbital angular momentum and $\bm{\hat{S}}$ is the electron spin. The atomic spin is defined as $\bm{\hat{F}}=\bm{\hat{I}}+\bm{\hat{J}}$. For a ground-state alkali or hydrogen atom, the electrons have no orbital angular momentum ($L=0$), so the electron’s total angular momentum quantum number is equal to its spin quantum number, that is $J=S=1/2$ [37]. In this scenario, if the atom is a boson, the nuclear spin must be half-integer, resulting in an integer atomic spin quantum number $F$. To realize the half-integer type of magnetic monopoles in these bosonic systems, we adopt the following two steps:

First, a strong magnetic field $\bm{B}_{0}=B_{0}{\bm{e}}_{z}$ with a strength of $B_{0}\gg AI/g\mu_{B}$ is applied to the system, so that the second term in Eq.(1) is much larger than the hyperfine interaction. Note that the nuclear magnetic moment is three orders of magnitude smaller than the electron magnetic moment. Thus, the second term of $\hat{H}_{\text{spin}}$ is dominant, and the angular momentum $\bm{J}$ is locked to the state $|J,-J\rangle_{{z}}$. In this manifold, the second term becomes $-g\mu_{B}B_{0}J$, and $A{\hat{I}_{z}}{\hat{J}_{z}}$ from the first term becomes $A\hat{I}_{z}/2$ (here, $J=1/2$ has been used). The other terms $A\hat{I}_{x}\hat{J}_{x}+A\hat{I}_{y}\hat{J}_{y}=A(\hat{I}_{+}\hat{J}_{-}+\hat{I}_{-}\hat{J}_{+})/2$ couples the high energy manifold $|J,J\rangle_{{z}}$, which will contribute a second order correction. Together with the last term, $-(\mu_{I}/I)B_{0}{\hat{I}_{z}}$, the low-energy effective Hamiltonian in the low-energy manifold $|J,-J\rangle_{{z}}$, up to the second order approximation, becomes

where $\omega_{L}=A/2+\mu_{I}B_{0}/I$. Using the fact that $J=1/2$, we have

Considering the condition $B_{0}\gg AI/g\mu_{B}$ and after discarding a constant term, we approximately have

which represents the energy splitting for the nuclear spin. This is different from the weak magnetic field case, where the low-energy physics is captured by the Zeeman splitting of the total spin $\bm{F}$ [19]. Note that we have neglected the second term (the second order correction) of Eq. (2) for simplicity, which contributes a linear $\propto\hat{I}_{z}$ and a quadratic term $\propto\hat{I}_{z}^{2}$. The linear term slightly corrects the magnitude of $\omega_{L}$ and does not change the form (4). The quadratic term can be compensated by the Floquet mechanics as shown below. For $\bm{I}=1/2$, which is our main focus for generating minimal monopoles, it is always a two-level system and the quadratic term becomes a constant term, so the second term of Eq. (2) just contributes a correction to the splitting $\omega_{L}$.

Second, similar to the proposal in Ref. [23], a periodic oscillating quadrupolar field

is applied. The quadratic term mentioned in above can be compensated by a proper choice of $\lambda_{0}$ [23]. Here we focus on $\lambda_{0}=1$. We require $\omega=\omega_{L}$ to couple different nuclear spin states resonantly. In the rotating frame of

the Hamiltonian becomes

The low-energy effective Hamiltonian in the rotating wave approximation [23], where $\omega_{L}\gg\mu_{I}B^{\prime}_{1}L/I$ being much larger than all other energy scales ($L$ is the typical size of the BEC and $B^{\prime}_{1}L$ charactorizes the magnitude of the quadrupolar magnetic field), becomes

When the two magnetic fields $\bm{B}_{0}$ and $\bm{B}_{1}$ are applied, the lowest spin state reads

where

is the common eigenstate of $\bm{\hat{I}}^{2}$ and $\bm{\hat{I}}\cdot\bm{e}_{r}$. Here, $(\theta,\phi)$ are the polar and azimuthal angles, respectively. In contrast to the integer $I$ case, for a half-integer $I$, the phase $e^{\pm iI\phi}$ is necessary to ensure $|I,-I\rangle_{{r}}^{\pm}$ remains the same value after the transformation $\phi\rightarrow\phi+2\pi$, where $\pm$ labels different gauge choices [23].

Now we consider the full single-particle Hamiltonian, which includes $H_{\text{spin}}$, the external trap potential ${V}_{H}(r)$, and the kinetic energy. It reads

where the external trap potential is ${V}_{H}(r)=m\omega^{2}_{H}\bm{{r}}^{2}/2$. Both magnetic fields, $\bm{B}_{0}$ and $\bm{B}_{1}$, are applied. In the adiabatic approximation of the lowest spin manifold (9), the effective Hamiltonian governing the dynamics of atoms in space is

where the vector potential is

and the scalar potential ${V}(r)={V}_{H}(r)-2\mu_{I}B^{\prime}_{1}r+I/2mr^{2}$ makes the BEC shell-shaped [23]. The reorientation of the spin state during the motion of the atom induces a gauge field and corrects the scalar potential. The effective magnetic field is $\bm{B}=\nabla\times\bm{A}=I{\bm{e}}_{r}/r^{2}$. This field exactly corresponds to an effective magnetic charge of $g_{m}=I$, which is half-integer in our systems.

In summary, to freeze the degree of freedom of the angular momentum of electrons, it requires a strong field $B_{0}\gg B_{c}\equiv AI/g\mu_{B}$, and to employ the rotating wave approximation, it requires the drive frequency to be much larger than other energy scales of the quadrupolar field, $\omega_{L}=A/2+\mu_{I}B_{0}/I\gg\omega_{c}\equiv\mu_{I}B^{\prime}_{1}L/I$. On the other hand, to use the adiabatic approximation for the lowest spin manifold, it is necessary to require the system’s temperature is much smaller than the energy splitting $\hbar\omega_{c}$. These three conditions are the key to realize a half-integer type of magnetic monopoles. In Table 1, we list the value of $B_{c}$ using the known experimental data, estimate the value of $\omega_{c}$ by assuming that the typical size of the BEC is $L=100~{}\mu$m and the magnetic field gradient is $B^{\prime}_{1}=0.5~{}\text{T/m}$, and evaluate $\omega_{L}$ by setting $B_{0}=1~{}\text{T}$, for different isotopes with a nuclear spin $I=1/2$, $3/2$, $5/2$, and $7/2$, respectively. The table shows that in this set, both conditions $B_{0}\gg B_{c}$ and $\omega_{L}\gg\omega_{c}$ are satisfied. Indeed, $\omega_{L}$ is of the order of $A/2$, which is always much larger than $\omega_{c}$. So $B_{0}\gg B_{c}$ is the only factor that should be taken into account. For hydrogen, $\hbar\omega_{c}\simeq 100~{}\text{nK}$, a uniform field with $B_{0}=0.25~{}\text{T}$ (about ten times of $B_{c}$) is sufficient to induce a minimal monopole of $g_{m}=1/2$. For ${}^{39}\text{K}$, ${}^{41}\text{K}$,${}^{85}\text{Rb}$,${}^{133}\text{Cs}$, $\hbar\omega_{c}$ is very small. Therefore, the required temperature is too low to be achievable in the current setting. A strategy to overcome this issue is to increase the radius of the shell-shaped BEC so that the energy splitting $\hbar\omega_{c}$ can be enhanced. Another concern is about gravitational effects, which may distort the shape of the BEC shell. A strategy to weaken this effect involves using optical trapping to counteract the gravitational potential or employing a microgravity environment.

In this section, we calculate the ground state of the spherical BEC in the presence of a minimal monopole with $I=1/2$. The dynamical behaviors of the BEC are described by the GPE,

where $\hat{H}_{0}=\hat{H}_{\pm}+\lambda_{3D}|\psi_{\pm}|^{2}$ contains the mean-field contribution from the contact atom-atom interaction, and the wavefunction is normalized to one in the whole space. To avoid Dirac strings—the singularities—from $\bm{A}_{\pm}$, the two gauges are utilized for $\psi_{\pm}$ in the regions

respectively, where $0<\delta<\frac{\pi}{2}$ [3]. We sketch this Wu-Yang approach in Fig. 1. In the overlap region, $R_{+}\cap R_{-}$, the wave functions are connected by the $U(1)$ gauge transformation

and accordingly, ${\bm{A}}_{+}={\bm{A}}_{-}-i\nabla\ln{S}$ [2]. Note the density distribution is gauge invariant, i.e., $|\psi_{+}|^{2}=|\psi_{-}|^{2}$.

We calculate the ground-state wave function when $l_{0}\equiv\mu_{I}B^{\prime}_{1}/m\omega_{H}^{2}\gg 1/\sqrt{m\omega_{H}}$, in which case the BEC is narrow shell-shaped with radius $r_{0}=2l_{0}$ and thickness $\sim 1/\sqrt{m\omega_{H}}$ [23]. For $\lambda_{3D}=0$, the ground state of $I=1/2$ can be constructed from the monopole harmonics [3], i.e.,

which vanishes at the vortex’s core position $(\theta_{v},\phi_{v})$ and its eigen-energy is $1/4mr_{0}^{2}$. On the other sector, $\psi_{-}$ is obtained via the gauge transformation: $\psi_{-}=S^{-1}\psi_{+}$.

In a closed manifold with monopoles inside, a $2\pi$ phase change of the wave function along a loop on the manifold does not always indicate a vortex, since it may be eliminated by a gauge transformation. Note that the velocity field is gauge invariant. Specifically for the case $\theta_{v}=\phi_{v}=0$, we have

The singularity of $\nabla\times\bm{v}$ indicates the position of the vortex. The winding number (i.e., the vortex charge) is given by the limit of the loop integral surrounding the vortex: $\lim_{l\rightarrow 0}({m}/{2\pi\hbar})\oint_{l}\bm{v}\cdot d\bm{r}=1$. Beyond the vortex center, the vorticity $\nabla\times\bm{v}=-\bm{B}/m$ is nonzero everywhere due to the presence of a monopole. The velocity field $\bm{v}$ cannot be expressed as the gradient of a scalar function with singular points, unlike in a system without monopoles [24].

Note that the system is symmetric under rotations with generators $\bm{\hat{L}}=\bm{r}\times\bm{\hat{p}}-I{\bm{r}}/r$, where $\bm{\hat{p}}=-i\nabla-\bm{{A}}_{\pm}$ [3]. The appearance of vortices breaks the rotational symmetry spontaneously.

For interacting cases, the ground state with a vortex locating at $\theta_{v}=\phi_{v}=0$, in general, has the form

and $\psi_{-}=S^{-1}\psi_{+}$. We will also use Dirac notation $\psi_{\pm}(\theta,\phi;\theta_{v},\phi_{v})\equiv\langle\theta,\phi|\psi_{\pm}(\theta_{v},\phi_{v})\rangle$. The velocity field is the same as Eq. (19), which is completely independent of the interaction strength. From GPE, we find that the amplitude $f(\theta)$ satisfies the gauge independent equation

where $\lambda=m\lambda_{2D}$ is dimensionless, $\lambda_{2D}$ is the effective two-dimensional interaction strength on the sphere (see Appendix A), and the normalization condition is $2\pi\int|f|^{2}\sin\theta d\theta=1$.

Approximately, we employ a trial solution $f(\theta)=a\tanh(b\theta)$ with fitting parameters $a$ and $b$. The fitting results agree very well with the numerical results of Eq. (21) especially for strong interaction cases, as shown in Fig. 2. For a small $\theta$, the square of the slope of $f(\theta)$, i.e., $a^{2}b^{2}$, linearly increases with $\lambda$, like the vortex solution in the plane case [37]. For large interactions, the density away from the vortex center becomes uniform and the vortex size is small. Consequently, $a^{2}\simeq 1/4\pi$ and $\mu\simeq\lambda/4\pi$ (in the Thomas-Fermi approximation). The numerical result of $\mu$ [see Fig. 2(b)] is a bit larger because the velocity fields are finite. For large $\lambda$, from Fig. 2(c), we have $b^{2}\simeq 0.051\lambda+0.76$ from the numerical fitting by replacing $a^{2}$ with $1/4\pi$. Therefore, the healing length $\theta_{c}=1/b$ decreases with the interaction strength. Correspondingly, the total particle number missing in the vortex decreases as well, and for $\theta_{c}\ll 1$, we have $n_{\text{miss}}\equiv 2\pi\int(a^{2}-[f(\theta)]^{2})\sin\theta d\theta\simeq 0.347\theta_{c}^{2}$.

The solution with the vortex core at an arbitrary position $(\theta_{v},\phi_{v})$ is obtained by the rotation transformation,

To engineer vortex at the target position in experiments, a small positive Gaussian potential with the center at $\bm{r}_{v}=r_{0}{\bm{n}}_{v}$ can be employed, where ${\bm{n}}_{v}=(\sin\theta_{v}\cos\phi_{v},\sin\theta_{v}\sin\phi_{v},\cos\theta_{v})$ is a unit vector. The potential breaks the rotation symmetry of the system explicitly, and the system has minimal total energy when the vortex center coincides with $\bm{r}_{v}$.

In the following, we calculate the expectation of the angular momentum, which will be employed in the study of the dynamics of the vortex. For the special case with the vortex center locating at the north pole, the expectation reads

where the angular momentum operator in the presence of a monopole is

Using the wave function (20), we have

Now we consider the three components $l_{i}=\bm{e}_{i}\cdot\bm{l}$ for $i=x,y,z$. Note that $\bm{e}_{x}\cdot\bm{e}_{\phi}=-\sin\phi$, $\bm{e}_{x}\cdot\bm{e}_{\theta}=\cos\theta\cos\phi$, and $\bm{e}_{x}\cdot\bm{e}_{r}=\sin\theta\cos\phi$. Therefore, $l_{x}$ vanishes since the integral over $\phi$ is null. For the $y$ component, we have $\bm{e}_{y}\cdot\bm{e}_{\phi}=\cos\phi$, $\bm{e}_{y}\cdot\bm{e}_{\theta}=\cos\theta\sin\phi$, and $\bm{e}_{y}\cdot\bm{e}_{r}=\sin\theta\sin\phi$. Similarly, we obtain $l_{y}=0$. For the $z$ component, we have $\bm{e}_{z}\cdot\bm{e}_{\phi}=0$, $\bm{e}_{z}\cdot\bm{e}_{\theta}=-\sin\theta$ , $\bm{e}_{z}\cdot\bm{e}_{r}=\cos\theta$. As a result,

Therefore, $\bm{l}=(0,0,1/2)$, which is the angular momentum for each atom. It is also easy to check that $|\psi_{+}(0,0)\rangle$ is indeed the eigenstate of $\hat{L}_{z}$ with eigenvalue $1/2$. We can generalize the expectation value of the angular momentum to the case with an arbitrary vortex position $(\theta_{v},\phi_{v})$ by a rotation transformation (22), which gives ${\bm{l}}\equiv\langle\hat{\bm{L}}\rangle=\bm{n}_{v}/2$. Note that the angular momentum is interaction-independent and its direction coincides with the vortex position vector.

In this section, we study the dynamics of the single vortex induced by the minimal monopole in an additional uniform magnetic field. Without loss of generality, we suppose that the magnetic field is along the $z$ direction, ${\bm{B}}_{2}=B_{2}{\bm{e}}_{z}$, and the corresponding vector potential is $\delta\bm{A}=B_{2}(-y/2,x/2,0)$. We assume that the monopole-vortex configuration is rigid. This is inspired by the Thiele equation for topological particles [40], such as magnetic skyrmions or Hopfions [41, 42]. These particles are typically treated as undeformed, particle-like entities that maintain their structure during motion, thereby simplifying their dynamics. Our assumption is also motivated by the analogy to vortex dynamics under strong interactions. When the healing length is small, a point vortex model is commonly employed to describe vortex motion [24], as phonon excitations are strongly suppressed in this regime. Our theoretical predictions, derived under this assumption, show excellent agreement with numerical results, thereby confirming its validity in the strong interaction region. For more general cases, where phonon excitations may deform the vortex profile and influence its trajectory, further research will be conducted to explore these effects.

Using this assumption, the evolving wavefunction becomes $|\psi(t)\rangle\propto|\psi_{\pm}[\theta_{v}(t),\phi_{v}(t)]\rangle$ with a varying vortex center. With this ansatz, the system is described by the dynamics of the vortex center, or equivalently, the dynamics of the angular momentum ${\bm{l}}(t)={\bm{n}}_{v}(t)/2$. From the GPE (14) and the definition $\bm{l}=\langle\psi(t)|\hat{\bm{L}}|\psi(t)\rangle$, we obtain

where the effective Hamiltonian

contains the gauge fields from both the monopole and the external magnetic field. We can split the effective Hamiltonian into two parts,

where

is the correction contributed by the external field. Using the fact that the instantaneous $|\psi(t)\rangle$ is the eigen-wavefunction of $\hat{H}_{\text{eff}}(\delta\bm{A}=0)$, we have

Note that $\nabla\cdot\delta\bm{A}=0$, $2\delta\bm{A}\cdot\hat{\bm{p}}=B_{2}[\hat{L}_{z}+(1/2)\cos\theta]$, and $\delta\bm{A}\cdot\delta\bm{A}=(1/4)B_{2}^{2}r^{2}\sin^{2}\theta$, the Hamiltonian becomes

where $\alpha=B_{2}/(2m)$ is the Larmor precession frequency and the scalar potential is

Detailed calculations give

where $\bm{\omega}=-\alpha\bm{e}_{z}$, and

If the corresponding density of the wave function $|\psi(t)\rangle$ is exactly uniform on the whole sphere, then we have $\langle\bm{e}_{\phi}\partial_{\theta}g(r,\theta)\rangle=0$ because $\bm{e}_{\phi}$ changes its direction along the circle of latitude. However, since the system has a small size of vacancy at the vortex center, and considering the shell-shaped case, $r\simeq r_{0}$, we have

Therefore,

where $\bm{\tau}=\partial_{\theta}g(r_{0},\theta)|_{\theta=\theta_{v}}n_{\text{miss}}\bm{e}_{\phi_{v}}$ is a torque contributed by the potential $g(r,\theta)$. In the following, we provide an interpretation of this result from a classical perspective. The effective energy for a vortex in the potential $g(r_{0},\theta)$ is $E(\theta_{v})=-g(r_{0},\theta_{v})n_{miss}$ due to the particle missing, then, the torque is $\bm{\tau}=\nabla_{\bm{l}}E(\theta_{v})\times\bm{l}$, which is exactly the same as the result given above. As a result, $l_{z}$ is conserved during the evolution. Therefore, $\theta_{v}$ remains a constant value, while the precession angular frequency is

where $\omega_{m}=-(1-n_{\text{miss}}){B_{2}}/{2m}+n_{\text{miss}}{B_{2}^{2}r_{0}^{2}\cos\theta_{v}}/{2m}$ arise from the magnetic field $B_{2}\hat{\bm{e}}_{z}$. The external magnetic field drives the precession of the vortex. $\omega_{m}$ is slightly different from the Larmor precession frequency due to the particle missing in the vortex. When considering the influence of a microgravity or an electric field along the $z$ direction, which gives an additional contribution in the scalar potential ($\propto\cos\theta$) in (33), the precession frequency will be further corrected.

In this section, we numerically simulate the vortex dynamics in the presence of $\bm{B}_{2}$ and compare it with the analytical results. For this purpose, we first develop the numerical approach. To avoid the singularities (Dirac strings) of the gauge field, we use the bi-gauge-evolution method to cover the whole sphere.

In conclusion, we have developed a pioneering method that enables the realization of magnetic monopoles with half-integer charges in the BEC systems. We discussed the suggested parameters for the experiment setup and explored the experimental flexibility. We have systemically studied ground-state vortex solutions and single-vortex dynamics on a sphere, which includes analytical calculations under the assumption of a rigid vortex-monopole structure and numerical simulations using a novel method to prevent singularities of the gauge fields. Larmor precession with a modified frequency captures the vortex dynamics on the sphere in the presence of a uniform electromagnetic field.

We note that the spherical BEC itself has garnered intense attention due to its nontrivial geometric properties. The bubble system realized in microgravity environments is also a platform to simulate cosmic inflation [45, 47, 46, 51, 50, 48, 49]. The nontrivial topology and the finite size endow phase transitions in spherical systems with novel characters [45, 29]. On a sphere, the vortex and antivortex always appear in pairs when there are no monopoles [52]. The existence of monopoles enables the investigation of vortex dynamics in a wider range of configurations.

Our research substantially advances the realization of magnetic monopoles, enhances our understanding of vortex dynamics and monopole properties, lays the foundation for experimental verification, and is helpful for further study of vortex dynamics in closed manifolds.