1. Introduction

In the Bose-Einstein condensate (BEC), the singular defect on the density is very common for producing the nontrivial quantum topological excitations, such as the monopole in the three dimensions [1–5] and the quantized vortex in the two dimensions [6,7]. These products are the fundamental interest for most of the physical researches. Unlike the scalar BEC, the spinor BEC opens a new dimensionality on the spin. It is found that the topological excitations, such as the half-quantum vortex and the Skyrmion, can be induced without density vanishing. Motivated by the normal exploration on the density singularity, we raise a question naturally: can we find the nontrivial topological excitations on the spin singularity? An obvious application of the possible spin singularity is in the field of the quantized message storage. Generally, the spin density (the absolute value |S|) is viewed as a conserved value. For example, we can prove that the absolute value |S| is fixed to be 1 for the spin-1/2 BECs. However, the spin density of the spin-1 BECs is not conserved due to the existence of the m_F = 0 component. This property is usually ignored in current investigations. It is unclear how to obtain some stable spin singular topological excitations till now.

Recent experiments of the spin-orbit coupling (SOC) BECs [8] reveal a new effect on bosons according to the artificial magnetic technology. And the SOC BECs are widely applied to explore the novel quantum phenomena [9–19]. Many investigations show the SOC BECs support the plane-wave phase [9], the stripe phase [9–12] and the half-quantum vortex state [11–14]. The combination effect of the SOC and the rotation on the spinor BECs has been shown to be able to generate various vortex structures [15,20,21]. Furthermore, the SOC has been shown to induce the half-skyrmion excitation in the spin-1 BECs [22, 23]. These impressive results enrich the quantum phenomena in the BEC system. Since both the SOC and the magnet field can manipulate the the spin of the BECs, we may wonder whether the combination of the SOC and the magnetic field could induce some spin singularity and stable nontrivial topological excitations without rotation.

In this paper, we will prove the spin singularity is a common phenomenon in the spin-1 BECs. Meanwhile, by combining the SOC and the magnetic field, our analysis proves that a singular spin topological excitation, whose wave function possesses property of the normal half-quantum vortex, can stably exist in the spin-1 BECs. In addition, the corresponding spin texture excitation of the singular spin topological excitation has the half-quantum topological charge. We find that the SOC plays a role in forming the local energy minimum to stabilize the topological excitation and that the magnetic field improves the occurrence of the spin singularity. Moreover, our mathematical proof shows an identical relation of the SOC energy of the BECs according to the position of the spin vortex.

The paper is organised as follows: In Sec. II, we analyse and show the singularity on spin in the spin-1 BEC. In Sec. III, we introduce our model and the assumption to fulfil the spin defect with singularity. In Sec. IV, we analyse the stability of the solution, prove the effect of the SOC and the magnetic field on the spin defect, and show the spin-interaction hardly affects the stability of this spin defect. Section V is a discussion about the results. A summary of the paper is presented in Sec. VI.

2. The spin-density singularity of the spin-1 Bose-Einstein condensates

For the spin-1 BECs, the spin vectors [24,25] is defined by

In Fig. 1, we display four examples of the spin density and the spin singularity. In Fig. 1(a), we assume the formation of the wave function to be Ψ = (sin(D), cos(D)sin(A), cos(D)cos(A))^T, where the two angles D and A can well characterize all the ratio of the three components and keep |Ψ|² = 1. The mauve dots demonstrate the emergence of the spin singularities (|S| = 0). Figure 1(a) shows that the spin singularities are very common in the spin-1 BEC even if there is no excitations. In Figs. 1(b), (c) and (d), we change the wave function to be Ψ = (sin(D), cos(D)sech[C|r−r₀|], cos(D) tanh[C|r−r₀|]e^iβ₀)^T. For β₀ = 0 or π, the appearance of the spin singularities covers all value of C|r − r₀|. For β₀ = π/2, the spin singularities only occur when the parameters are C|r − r₀| = 0 and D = nπ, where n is integer. In fact, we find that sin(D) = 0, C|r − r₀| = 0 are the common condition to create the spin singularities for different β₀. Therefore, the spin singularity is a common phenomenon in the spin-1 BECs.

 

Fig. 1 Several schematics of the spin density and the spin singularity in the spin-1 BECs. (a) The formation of wave function is Ψ = (sin(D), cos(D)sin(A), cos(D)cos(A))^T. In (b), (c) and (d), the formation of the wave function is Ψ = (sin(D), cos(D)sech[C|r − r₀|], cos(D) tanh[C|r − r₀|]e^iβ₀)^T. (b) β₀ = 0, (c) β₀ = π/2 and (d) β₀ = π. The color denotes the value of |S| and the mauve dots indicate the positions of the spin singularities.

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3. Model

In the spin-1 BECs, to force one of the components approaching zero, i.e., Ψ_j(j = 0, ±1) → 0, we can induce the vortices whose center is a singularity. Furthermore, we can also magnetize the BECs. This idea may produce a scalar BEC with only one component. While the SOC technology may change the result partly. In this study, we consider the combination of the SOC and the magnetic field. The dynamical equations of the system are written as

In a scalar BEC, we can set vortices by having the ground state wave function multiplied the ansatz function [26]

4. Analysis on stability

In the calculations of this Section, we use ħ = 1, the atomic mass m = 1, and the chemical potential μ = 1ħω. Meanwhile, we set the interaction coefficient g_n = 1 and ω = 0.1c/ξ for the external potentials [$V(r)=\frac{1}{2}m{\omega }^2{r}^2$], where $c=\sqrt{\mu /m}$is the Bogoliubov speed of sound and $\xi =\hslash /\sqrt{\mu m}$ is the healing length. Thus, the length unit is ξ, the energy unit is μ and the time unit is ξ/c. The unit of B_⊥ and B_z is ħω. The unit of κ_x and κ_y is $\sqrt{\mu /m}$. And the unit of g_n and g_s is ξ³μ. Note that we only indicate the parameter values without showing the corresponding units in the following part of this Section.

In the numerical calculations, we set the size of the system from −15ξ to 15ξ, where the size is much larger than that of the BECs. Then, we set the vortex with Eq. 7. We obtain n₀ through Thomas-Fermi approximation n₀(r) = [μ−V(r)]/g_n. In two spatial dimensions, the wavefunction is discretised onto a linear grid with separation Δx (Δy) with index i (j), expressed as Ψ_n(x_i, y_j), n = 0, ±1. The central space-difference operator is given by $\frac{\partial {\mathrm{\Psi }}_n(x_i,y_j)}{\partial x}=\frac{{\mathrm{\Psi }}_n(x_{i+1},y_j)-{\mathrm{\Psi }}_n(x_{i-1},y_j)}{2\mathrm{\Delta }x}$, $\frac{\partial {\mathrm{\Psi }}_n(x_i,y_j)}{\partial y}=\frac{{\mathrm{\Psi }}_n(x_i,y_{j+1})-{\mathrm{\Psi }}_n(x_i,y_{j-1})}{2\mathrm{\Delta }y}$. Then the double space derivative operator is given by $\frac{{\partial }^2{\mathrm{\Psi }}_n(x_i,y_j)}{\partial x^2}=\frac{{\mathrm{\Psi }}_n(x_{i+1},y_j)+{\mathrm{\Psi }}_n(x_{i-1},y_j)-2{\mathrm{\Psi }}_n(x_i,y_j)}{\mathrm{\Delta }{x}^2}$ and $\frac{{\partial }^2{\mathrm{\Psi }}_n(x_i,y_j)}{\partial y^2}=\frac{{\mathrm{\Psi }}_n(x_i,y_{j+1})+{\mathrm{\Psi }}_n(x_i,y_{j-1})-2{\mathrm{\Psi }}_n(x_i,y_j)}{\mathrm{\Delta }{y}^2}$. Using Eq. 5, we can obtain the total energy of the system.

4.1. The stability of the solution

Since Eq. 7 has assumed an approximate half-quantum vortex solution, we now check whether this solution is possible to occur in the spin-1 BECs. Using Eq. 5 of the dimensionless formation, we check the energy of the BECs. Firstly, we consider the effect of ratio of the |F = 1, m_F = 1〉 component on the energy of the BECs. Figure 2(a) shows the curves of the energy of the BECs as the parameter C changes. When |sin(D)| = 0, the energy curve locates at the bottom. As the |F = 1, m_F = 1〉 component increases, i.e., |sin(D)| grows, the energy curve shifts up. The black arrow indicates the trend. Thus, to minimize the energy of the BECs, we should set |sin(D)| = 0, i.e., the |F = 1, m_F = 1〉 component vanishes in the spin-1 BECs.

 

Fig. 2 Energy of the spin-1 BECs with a possible half-quantum vortex. (a) The effect of |F = 1, m_F = 1〉 component on the energy of the BECs as the variational parameter C changes. Other parameters are g_n = 1, g_s = 0, ω = 0.1, B_z = 0.5, B_⊥ = 0, κ_x = κ_y = κ = 1 and (x₀, y₀) = (0, −0.5). (b) The effect of the SOC on the energy of the BECs as the variational parameter C varies. (c) The effect of the B_z on the energy of the BECs as the variational parameter C varies. (d) The effect of the B_⊥ on the energy of the BECs as the variational parameter C varies. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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In Fig. 2(b), we set the magnetic field B_z = 0.5. For κ = 0, the energy of BECs decreases monotonously as C increases. This implies that the energy of BECs will be the minimum when C → ∞, i.e., the size of vortex approaches 0. Thus, we can not get stable vortex with finite size when κ = 0. When κ is more than 0, there is a local dip of the energy of BECs. The larger the value of κ is, the more obvious the dip is. The dip indicates that we can obtain the minimum energy by using Eq. 7 with the corresponding C^E_min. Furthermore, the vortex is supposed to be stable. Figure 2(b) implies that the SOC is a necessary factor for the stable half-quantum vortex of Eq. 7.

In Fig. 2(c), we further check the effect of the magnetic field B_z and C on the energy of the BECs. With κ = 1, the energy dip occurs in every case of B_z. And the large B_z causes a relative low energy. This phenomenon could be easily understood. The main reason is that the magnetic field is effective on the |F = 1, m_F = −1〉 component. Figure 2(d) indicates the effect of B_⊥ on the energy of the BECs when the system possesses a half-quantum vortex of Eq. 7. Generally speaking, B_⊥ does not lead to an obvious change of the energy curves. However, B_⊥ can cause the transformation between the |F = 1, m_F = 0〉 component and the |F = 1, m_F = 1〉 component. This implies the increase of |sin(D)|. Therefore, the magnetic field of B_⊥ should be set to be 0 for the stable half-quantum vortex.

Figures 3(a)–(d) show an example of the half-quantum vortex of C^E_min = 0.38. Figures 3(a) and (c) are the densities of the |F = 1, m_F = 0〉 and |F = 1, m_F = −1〉 components respectively, and Figures 3(b) and (d) are the corresponding phases. Figure 3(e) is the density profile of the vortex. For purposes of comparison, we plot the density profile of a normal vortex of Eq. 6 with a homogenous background density. Figure 3(f) shows the corresponding spin density |S| and Figure 3(g) further indicates the profile of |S|. Obviously, the spin density |S| decreases to be 0 at the center of vortex.

 

Fig. 3 Approximate half-quantum vortex solution in the spin-1 BEC and the corresponding singular spin texture. (a),(b),(c) and (d) show an example of the half-quantum vortex. The parameters are g_n = 1, g_s = 0, ω = 0.1, sin(D) = 0, C = 0.38, B_z = 0.5, B_⊥ = 0, κ_x = κ_y = κ = 1 and (x₀, y₀) = (0, −0.5). (a) and (b) are the densities of the |F = 1, m_F = 0〉 and |F = 1, m_F = −1〉 components, respectively. (c) and (d) are the corresponding phases. (e) shows the profile of the half-quantum vortex. The green curve is the profile of a normal vortex with Eq. 6. (f) Spin density |S|. (g) The profile of the spin density |S|. (h) Spin texture. (i) Topological charge density q(x, y). The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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We plot the spin texture in Fig. 3(h) with the spin vector (S_x, S_y, S_z). Far away from the centre of the spin vortex, all the arrows are perpendicular to the x − y plane, pointing into the −z direction. While approaching the coordinate (0, −0.5), the arrows become parallel to the x − y plane and the arrows shorten to be zero (i.e., |S| → 0). Meanwhile, the arrows form a circle around (0, −0.5). These properties show that it is a special topological excitation. Figure 3(i) indicates the topological charge density q(x, y), which is defined by $q(x,y)=\frac{1}{4\pi }\mathbf{s}\cdot \left(\frac{\partial \mathbf{s}}{\partial x}\times \frac{\partial \mathbf{s}}{\partial y}\right)$, where s = S/|S|. We find that there is a singular point at (0, −0.5) and that the sum of the topological density is Q = ∬ q(x, y)dxdy = 0.5. Therefore, we predict a half-quantum vortex in the spin-1 BECs and this vortex is related to a half-quantum topological spin texture with singularity.

The phase of spin defect can be written as ${\theta }_{\mathit{spin}}=\text{arctan}\left(\frac{S_y}{S_x}\right)$. Using Eq. 7 with sin(D) = 0, we can obtain θ_spin = θ.

We now check the effect of the position of the vortex on the energy of the BECs. In Fig. 4(a), we use the variational method to examine the minimum energy E_min(x, y). Note that we set the half-quantum vortex at position (x, y). Around (0, 0⁻), we can find a local energy minimum, which can be viewed as the most stable position (x_s, y_s). This implies that it is possible to obtain a stable vortex in the spin-1 BECs when we quench the BECs. If the local energy minimum is not around the centre of BEC but approaching the boundary of the BECs, the vortex will be drifted out of the BECs though we firstly fix it at the centre of the BECs. In this case, the half-quantum vortex of Eq. 7 is in an unstable state [see Fig. 4(c), where the most stable position (x_s, y_s) approaches the boundary of the system]. Figure 4(b) indicates the corresponding parameter C^E_min (x, y), in which the whole BECs with the half-quantum vortex have the minimum energy. Thus, the vortex of C^E_min (x_s, y_s) possesses the minimum energy and should be the most possible solution in this system.

 

Fig. 4 (a) The stable minimum energy E_min of the BECs as a function of the position of the vortex (x, y). We set B_z = 0.5 and κ_x = κ_y = κ = 1. (b) The corresponding parameter C^E_min as a function of the position of the vortex (x, y). (c) The unstable minimum energy E_min of the BECs as a function of the position of the vortex (x, y). We set B_z = 1.5 and κ = 0.5. (d) Stability of the half-quantum vortex under the magnetic field B_z and the SOC κ. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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Obviously, each pair of parameters of (κ, B_z) can cause a most stable position (x_s, y_s) in the x − y plane. If all the most stable positions (x_s, y_s) could be found out, we will be able to obtain an effective phase diagram, which shows the stable position and the unstable position in the κ − B_z plane when the spin vortex is set. For reducing the amount of calculation, we can choose a finite region $r(x,y)=\sqrt{x^2+y^2}<10$, where the density n₀(r = 10) is very low. The main work now is to check the formation of the minimum energy E_min(x, y) of the BECs, where (x, y) is the position of the spin vortex. If E_min(x, y) behaves like Fig. 4(a), the spin vortex would be stable in the most stable position (x_s, y_s). If E_min(x, y) behaves like Fig. 4(c), the spin vortex would be unstable in the system. Figure 4(d) plots $\sqrt{x_s^2+y_s^2}$ of the most stable position (x_s, y_s) for all the cases of (κ, B_z). On the other hand, Figure 4(d) also shows the phase diagram of stability of the half-quantum vortex in the spin-1 BECs. If the value of $\sqrt{x_s^2+y_s^2}$ is less than 10, we view the vortex to be stable in this system. Reversely, the vortex would be unstable and go to the boundary in this system. In fact, we can not obtain the stable half-quantum vortex in the unstable region, because E_min(x, y) has the formation like Fig. 4(c). And the so-called most stable position (x_s, y_s) and the corresponding $\sqrt{x_s^2+y_s^2}=10$ is false. Thus, we do not give a scale in Fig. 4(d) to avoid the value to cause some misunderstanding.

Why does the position of the minimum energy approach (0, 0⁻) in Fig. 4(a)? To clarify this phenomenon, we examine the energy of the BECs when we set the position of the vortex to be (x₀ = 0, y₀ = Y), and the size of the vortex is fixed with C = 0.4. Figure 5(a) shows the total energy. Obviously, when κ = 0, we can find that the total energy is continuous as Y varies. But the energy approaches the maximum value around (0, 0), i.e., the vortex would be unstable when it locates at the centre of BECs. With κ > 0, the total energy is uncontinuous as Y varies. There is a step at Y = 0. In Fig. 5(b), we can find E_SOC = 0 when κ = 0. When κ > 0 and Y < 0, we have E_SOC < 0. When κ > 0 and Y > 0, we have E_SOC > 0. Thus, the SOC causes the energy break when the position of the vortex (x₀ = 0, y₀ = Y) changes. However, the minimum of the total energy is around the centre only when κ ≥ 0.6 in Fig. 5(a).

 

Fig. 5 The SOC causes the energy break as the position of the vortex changes. The strength of the SOC is κ_x = κ_y = κ. (a) The total energy of the system as the position of the vortex (x₀ = 0, y₀ = Y) varies. (b) The corresponding energy of the SOC term, i.e., E_SOC. We set the half-quantum vortex with C = 0.4 in the BECs, and strength of the magnetic field B_z = 0.5. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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In the above text, the position of the minimum energy is (0, 0⁻), which is caused by the SOC. Note that (0, 0) is a singular position for the half-quantum vortex of Eq. 7 and there is no corresponding image vortex. In the discussion (see Sec. V), we give a detail description of the image vortex. Thus, we can not set the vortex at (0, 0⁻) directly but set it with a little deviation in the calculation. Here, the deviated position is (0, −0.5).

Figure 6(a) further shows the most stable position |r| ($\left|r\right|=\sqrt{x_s^2+y_s^2}$) changes as the SOC κ varies. For the weak SOC, vortex tends to be drifted out of the BECs, i.e., the value of |r| is up to 10 and we point out it is unstable. While the strong SOC can stabilize the vortex in the centre of the BECs, i.e., the value of |r| approaches 0. Figure 6(b) indicates the effect of the SOC on C^E_min (x_s, y_s). Generally speaking, the strong SOC mainly decreases the values of C^E_min (x_s, y_s) while the strong magnetic field increases the values of C^E_min (x_s, y_s). Figure 6(c) further shows the most stable position |r| ($\left|r\right|=\sqrt{x_s^2+y_s^2}$) changes as the magnetic field B_z varies. In the absence of the SOC, the vortex would be drifted out of the BECs, and the magnetic field can not cause an effective stable state. We find that the strong magnetic field B_z would cause the value of |r| to increase. This point is obvious for the weak SOC (κ = 0.5 and 1). For the strong SOC (κ = 1.5 and 2), the vortex also should be drifted out of the BECs if B_z is strong enough. Figure 6(d) indicates that C^E_min (x_s, y_s) increases as B_z goes up. With the weak SOC, the value of C^E_min (x_s, y_s) grows more quickly but the vortex is easy to reach the unstable state.

 

Fig. 6 (a) The most stable position of the half-quantum vortex with the parameter C^E_min in the BECs as the SOC κ (κ_x = κ_y = κ) varies. $\left|r\right|=\sqrt{x_s^2+y_s^2}$. (b) The corresponding parameter C^E_min as a function of κ. (c) The most stable position of the half-quantum vortex with C^E_min in the BECs as the magnetic field B_z varies. (d) The corresponding parameter C^E_min as a function of B_z. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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4.2. The effect of the spin-orbit coupling on the stability

We find that the SOC plays a role in stabilizing the vortex in the BECs. In Fig. 7(a), the strength of the SOC is 0, so e_SOC is 0 naturally. If κ > 0, there is a local minimum value of e_SOC(x, y), which is less than 0 and just at the position of the vortex [see Fig. 7(b)]. This local minimum value stabilizes the vortex in the BECs. The stronger the SOC is, the more obvious the local minimum of e_SOC(x, y) is [see Fig. 7(c)]. In Fig. 7(d), we further examine the minimum value of e_SOC(x, y), i.e., $e_{\mathit{SOC}}^{\mathit{min}}$. It decreases as the SOC increases. Similarly, the whole energy E_SOC coming from the SOC also decreases as the SOC increases [see Fig. 7(e)].

 

Fig. 7 The effect of the SOC on the SOC energy e_SOC(x, y) of the BECs under the parameter C^E_min (0, −0.5). (a) κ_x = κ_y = κ = 0. (b) κ_x = κ_y = κ = 0.5. (c) κ_x = κ_y = κ = 1. (d) The minimum values of e_SOC as a function of the SOC κ. (e) The sum of the SOC energy E_SOC [E_SOC = ∬ e_SOC(x, y)dxdy] as a function of the SOC κ. We set B_z = 0.5 and the position of the vortex to be (0, −0.5). The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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4.3. The effect of the magnetic field on the stability

Figure 8 shows the effect of the magnetic field B_z on the SOC energy of the BECs. In Figs. 8(a)–(c), we can find a local minimum value of e_SOC(x, y). Meanwhile, comparing the three cases in Figs. 8(a)–(c), we find that the magnetic field B_z would enhance the effect of the local minimum e_SOC(x, y) and make it more obvious. Figure 8(d) further shows that the local minimum value of e_SOC(x, y) is less than 0 and the magnetic field B_z causes its value to decrease. In fact, if we examine the total SOC energy E_SOC, we find that it increases as the SOC goes up [see Fig. 8(e)].

 

Fig. 8 The effect of the magnetic field B_z on the SOC energy e_SOC(x, y) of the BECs under the parameter C^E_min (0, −0.5). (a) B_z = 0. (b) B_z = 0.5. (c) B_z = 1. (d) The minimum values of e_SOC as a function of the magnetic field B_z. (e) The sum of the SOC energy E_SOC as a function of the magnetic field B_z. We set the SOC κ_x = κ_y = κ = 0.5 and the position of the vortex to be (0, −0.5). The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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4.4. The effect of the spin interaction g_s on the stability

The spin interaction g_s normally plays a key role in determining the properties of the spin-1 BECs. In the above text, we have assumed g_s = 0, i.e., a₂ = a₀ [$g_s=\frac{4\pi {\hslash }^2(a_2-a_0)}{3m}$]. We now consider the effect of g_s. According to the previous studies, we know a₂ ≈ a₀. For example, the two parameters are a₀ = 101.8a_B and a₂ = 100.4a_B in the spin-1 BEC of ⁸⁷Rb, and are a₀ = 50a_B and a₂ = 55a_B in the spin-1 BEC of ²³Na. Thus, we can assume g_s ≪ g_n.

Since the SOC and the magnetic field B_z are the two key points in our above analysis, we can assume that the spin interaction g_s would appear in the following four cases: (i) both the SOC and the magnetic field B_z are considered, (ii) only the SOC exists, (iii) both the SOC and the magnetic field B_z are turned off, and (iv) only the magnetic field B_z exists.

We have found that (0, 0⁻) is the stablest position if the half-quantum vortex can exist in the BEC stably. Firstly we test the energy of the BECs with the half-quantum vortex locating at (0, −0.5). In Fig. 9(a), we set κ_x = κ_y = κ = 1 and B_z = 0.5 and examine the total energy of the BECs. We can see the lowest energy with suitable C. We have marked them with stars. The lowest energy with C > 0 implies the possible stable solution of the half-quantum vortex in the BECs. Similarly, Figure 9(b) is the result of the case (ii), in which we set κ_x = κ_y = κ = 1 and B_z = 0. The lowest energy with suitable C also exists (see the stars), no matter what the value of g_s is. Figure 9(c) shows the result of the case (iii) with κ_x = κ_y = κ = 0 and B_z = 0. Obviously, when g_s > 0, the lowest energy approaches C = 0. When g_s < 0, we have the lowest energy with C > 0. This property indicates the half-quantum vortex is not stable in this system with g_s > 0. Thus, the ferromagnetic BECs (g_s < 0) is beneficial to keep the half-quantum vortex. For the case (iv), we only set the magnetic field B_z = 0.5 and turn off the SOC, i.e., κ = 0. In Fig. 9(d), we can also find the lowest energy with C > 0.

 

Fig. 9 The total energy of BECs as g_s and C change. We set the position of the vortex to be (0, −0.5). (a) κ_x = κ_y = κ = 1 and B_z = 0.5. (b) κ_x = κ_y = κ = 1 and B_z = 0. (c) κ_x = κ_y = κ = 0 and B_z = 0. (d) κ_x = κ_y = κ = 0 and B_z = 0.5. The stars indicate the best C to create the lowest energy as g_s varies. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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Whether it is true that the lowest energy with C > 0 can keep the solution stable in Fig. 9? We now open the degree of freedom to let the vortex locate at different positions, and compare the total energy of the BECs to examine whether the lowest energy can appear around (0, 0⁻). In Fig. 10, we show the total energy of the BECs when we change the position of the vortex to (x₀ = 0, y₀ = Y). Figure 10(a) is the result of the case (i) with g_s = 0.2, which stands for a strong antiferromagnet. We indeed find that the lowest energy locates around (0, 0⁻). This result indicates that g_s does not affect the stability of the half-quantum vortex in Fig. 9(a). In Fig. 10(b), we also can find that the stablest region is around (0, 0⁻). However, we must point out that the half-quantum vortex of Eq. 7 can not appear with sin(D) = 0. Because the SOC would cause atoms exchange among the m_F = 0 and m_F = +1 component, it is hard to keep the m_F = +1 component approaching 0 without the magnet field B_z. In the previous studies, it has been proven that the pure SOC creates the plane-wave phase and the stripe phase in BECs. Therefore, the stable vortex can not be provided in the case (ii).

 

Fig. 10 The total energy of BECs as the position of the vortex (x₀ = 0, y₀ = Y) and C change. (a) g_s = 0.2, κ_x = κ_y = κ = 1 and B_z = 0.5. (b) g_s = 0.2, κ_x = κ_y = κ = 1 and B_z = 0. (c) g_s = −0.2, κ_x = κ_y = κ = 0 and B_z = 0. (d) g_s = −0.2, κ_x = κ_y = κ = 0 and B_z = 0.5. The arrow indicates the stablest region, where the energy is the lowest. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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In Fig. 10(c), we choose the strong ferromagnet BECs of g_s = −0.2. Our calculation indicates that the lowest energy region is not around (0, 0⁻) but around the boundary of the BECs (see the black arrow). Thus, we can not obtain the stable half-quantum vortex without the SOC or the magnetic field B_z. Similarly, only with the magnetic field B_z, the lowest energy region also approaches the boundary of the BECs in Fig. 10(d).

Generally speaking, with both the SOC and the magnetic filed B_z, the half-quantum vortex can stably exist in the BECs. The adjustment of the spin interaction g_s could hardly affect the stability of the half-quantum vortex.

5. Discussion

In this work, we have studied the stability of the half-quantum spin vortex in the spin-1 SOC BECs, which are under the magnetic field. Especially in Fig. 4(a), the minimum energy appears in the region of y < 0. In the discussion, we firstly discuss whether the half-quantum spin vortex is stable only in y < 0. In fact, the spin vortex can also be stable in the region of y > 0 when κ < 0. In Fig. 11, we provide a case with κ = −1 and all other parameters are as the same as those in Fig. 4(a). Obviously, the stable point, i.e., the position of the minimum energy, is in the region of y > 0. This case shows that the stable point of the spin vortex is affected by the SOC. And it is the SOC that causes the spin vortex to be deflected into a side of the x − y plane.

 

Fig. 11 The minimum energy E_min of the BECs as a function of the position of the vortex (x, y). We set B_z = 0.5 and κ_x = κ_y = κ = −1. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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Secondly, we illuminate a detail about setting the vortex in the BECs. Ref. [26] provides a method to set vortex in the BEC. In fact, when applying the method, we need to set the corresponding image vortex, which has the reverse winding number outside of the BECs. Meanwhile, the vortex phase cut between −π and π can extend fastest to the outskirts of the BECs when it connects the image vortex. The image vortex of the vortex is located at $r_0^{\prime }=\frac{R^2{r}_0}{{\left|r_0\right|}^2}$, where R is the radius of the BECs and r′₀ = (x′₀, y′₀). The corresponding phase is ${\beta }^{\prime }(x,y)=\text{arctan}\left(\frac{y-y_0^{\prime }}{x-x_0^{\prime }}\right)$. In this study, we set R = 15ξ which is a little larger than the size of the BECs and it can make the image vortex locate at the outside of the BECs. In Figs. 12(a1) and (b1), we display the phases with the vortex at the position of (0, −0.5) and (0, 0.5) respectively. Because the density outside of the BECs is set to be 0, the image vortex can not be seen. But it changes the phases of the BECs. If we do not consider the image vortex, the phase cut between −π and π would always parallel to the x axis.

 

Fig. 12 (a1)–(a4) are the results of the spin vortex with C = 0.4 at (0, −0.5). (a1) phase of ${\mathrm{\Psi }}_{-1}^{(0,-0.5)}$. (a2) the real part of ${\partial }_x{\mathrm{\Psi }}_{-1}^{(0,-0.5)}(x,y)$. (a3) the imaginary part of ${\partial }_x{\mathrm{\Psi }}_{-1}^{(0,-0.5)}(x,y)$. (a4) $e_{\mathit{soc}}^{(0,-0.5)}(x,y)$. (b1)–(b4) are the results of the spin vortex with C = 0.4 at (0, 0.5). (b1) phase of ${\mathrm{\Psi }}_{-1}^{(0,0.5)}$. (b2) the real part of ${\partial }_x{\mathrm{\Psi }}_{-1}^{(0,0.5)}(x,y)$. (a3) the imaginary part of ${\partial }_x{\mathrm{\Psi }}_{-1}^{(0,0.5)}(x,y)$. (a4) $e_{\mathit{soc}}^{(0,0.5)}(x,y)$. For all other parameters, we use B_z = 0.5, B_⊥ = 0 and κ_x = κ_y = κ = 1. The energy, magnetisation, length, strength of the SOC and strength of the interaction are in units of ħω, ħω, ξ, $\sqrt{\mu /m}$, and ξ³μ, respectively.

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Thirdly, we give a proof to further indicate how the SOC causes the vortex to deflect to one side of the x − y plane. Take the two cases with spin vortex at the position of (0, −0.5) and (0, 0.5) as an example. Certainly, the size of the spin vortex in the two cases is the same, i.e. C is fixed and we only change the position of the spin vortex. In fact, we can obtain ${\mathrm{\Psi }}_1^{(0,-0.5)}(x,y)={\mathrm{\Psi }}_1^{(0,0.5)}(x,y)=0$, ${\mathrm{\Psi }}_0^{(0,-0.5)}(x,y)={\mathrm{\Psi }}_0^{(0,0.5)}(-x,-y)\ge 0$ and ${\mathrm{\Psi }}_1^{(0,-0.5)}(x,y)={\mathrm{\Psi }}_{-1}^{(0,0.5)}(-x,-y)$. The superscripts (0, −0.5) and (0, 0.5) denote the position of the vortex. Meanwhile, we have ${\partial }_x{\mathrm{\Psi }}_{-1}^{(0,-0.5)}(x,y)=-{\partial }_x{\mathrm{\Psi }}_{-1}^{(0,0.5)}(-x,-y)$ and ${\partial }_y{\mathrm{\Psi }}_{-1}^{(0,-0.5)}(x,y)=-{\partial }_y{\mathrm{\Psi }}_{-1}^{(0,0.5)}(-x,-y)$. For clearly displaying these features, Figures 12(a2) and (a3) plot the real part and imaginary part of ${\partial }_x{\mathrm{\Psi }}_{-1}^{(0,-0.5)}(x,y)$, respectively. In addition, Figures 12(b2) and (b3) plot the real part and the imaginary part of ${\partial }_x{\mathrm{\Psi }}_{-1}^{(0,0.5)}(x,y)$, respectively. Thus, the SOC energy can be written as

6. Conclusion

We have reported a new type of half-quantum topological excitation whose singularity is on the spin in the spin-1 BECs. Our numerical calculations show that this kind of excitation is stable under the combined restriction of the SOC and the z-direction magnetic field. The spin interaction g_s hardly change the stability of this kind of excitation. We not only show the formation of the approximate solution, but also indicate the most suitable coefficient C^E_min for the solution under various conditions. Furthermore, we prove the identical relation of the SOC energy $E_{\mathit{soc}}^{(x_0,y_0)}=-E_{\mathit{soc}}^{(-x_0,-y_0)}$, where (x₀, y₀) is the position of the spin vortex. This study indicates a brand new vortex with defect on the spin and the stability mechanism. It is of particular significance for exploring such topological excitations in other quantum gas and condensed matter physics.