Spherical Wave Half Vortex (SWHV)

• Spherical Wave Half Vortex (SWHV) is a condensate state in two-component Bose gases characterized by a radial phase factor e^(ik_c r) and a meron spin texture carrying a half-integer topological charge.
• It distinguishes itself from the ordinary half vortex by exhibiting a finite spherical-wave momentum, resulting in dual ring peaks in momentum space detectable through interferometric techniques.
• Variational analysis and Bogoliubov-de Gennes stability studies confirm that the SWHV is a robust, experimentally accessible phase in systems with Rashba spin-orbit coupling and anisotropic interactions.

The spherical wave half vortex (SWHV) is a topological condensate state identified in a two-component Bose gas with Rashba spin-orbit couplings. In the classification introduced by Chen et al., it is one of four topological structures found in the system, alongside the half vortex (HV), double peak (DP), and spin spiral (SS) phases; the SWHV differs from the ordinary HV by the presence of a radial phase factor $e^{i k_c r}$ in the condensate wavefunction, while retaining a meron spin texture with half-integer topological charge (Chen et al., 4 Sep 2025).

1. Defining condensate form

In polar coordinates $(r,\theta)$, the SWHV with total angular momentum $j_z=\tfrac12$ is written as

$$\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$$

where $f(r),g(r)\ge 0$ are real radial profiles satisfying

$\int d^2r\,[f^2+g^2]=1,$

and $k_c$ is the spherical-wave momentum determined variationally in units $a_\perp^{-1}$ (Chen et al., 4 Sep 2025).

This representation isolates the feature that distinguishes the SWHV from the ordinary half vortex: when $k_c=0$, the state reduces to the HV. The SWHV is therefore a finite-momentum half-vortex state rather than a distinct winding sector. A plausible implication is that the relevant distinction is not the componentwise vorticity pattern, which remains that of a half vortex, but the radial phase structure and the associated momentum-space response.

2. Spinor structure and meron mapping

The spinor components exhibit unequal phase windings: $$\psi_\uparrow=f(r),\qquad \psi_\downarrow=g(r)e^{i\theta}.$$ Thus, $(r,\theta)$0 carries no angular phase, whereas $(r,\theta)$1 carries a $(r,\theta)$2 vortex (Chen et al., 4 Sep 2025).

The local Bloch vector is

$(r,\theta)$3

with

$(r,\theta)$4

Within this parametrization, the SWHV is mapped to a meron texture. At $(r,\theta)$5, $(r,\theta)$6, implying $(r,\theta)$7. At $(r,\theta)$8, $(r,\theta)$9, so $j_z=\tfrac12$0 and $j_z=\tfrac12$1; the spins then lie in-plane and wind once during a full $j_z=\tfrac12$2 circuit in $j_z=\tfrac12$3, while covering only half the Bloch sphere. The same summary identifies both HV and SWHV as merons, indicating that the radial spherical-wave factor does not alter the local meron mapping itself (Chen et al., 4 Sep 2025).

3. Half-integer topology

The topological characterization is given by the skyrmion-type charge

$j_z=\tfrac12$4

for which the SWHV has

$j_z=\tfrac12$5

The half-integer value is obtained by substituting the above $j_z=\tfrac12$6 into the topological density in polar coordinates (Chen et al., 4 Sep 2025).

This identifies the SWHV as a half-skyrmion in the spin sector. The terminology used in the source material is therefore precise: the SWHV is a half vortex at the level of condensate phase structure and simultaneously a meron at the level of spin texture. A common misconception is that the additional factor $j_z=\tfrac12$7 changes the topological charge; in the description given here, it does not. The half-integer topology is inherited from the Bloch-vector texture rather than from the radial phase factor.

4. Variational energetics and phase selection

In trap units, with $j_z=\tfrac12$8 and $j_z=\tfrac12$9, the mean-field energy of the $$\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$$0 ansatz is \begin{align} E[\phi_0] &=N!\int d^2r\;\Bigl{ \tfrac1{2m}\bigl[\,(f')^2+(g')^2 +k^2f^2+(k^2+\tfrac1{r^2})g^2 -4\,k\,k_0\,f\,g\bigr]\nonumber$\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$1 whereas the SWHV is

$\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$2

The difference is therefore entirely encoded in the radial phase factor (Chen et al., 4 Sep 2025).

Property	HV	SWHV
Wavefunction	$\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$3	$\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$4
In-situ densities	$\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$5	Density unchanged
Momentum-space structure	Peak at $\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$6	Two ring peaks at $\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$7

The source summary emphasizes three consequences of this comparison. First, HV and SWHV have identical in-situ densities $\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$8 and $\Psi_{\rm SWHV}(r,\theta) = \phi_0(r,\theta) = e^{\,i k_c r}\, \begin{pmatrix} f(r) \ g(r)\,e^{\,i\theta} \end{pmatrix},$9. Second, the local spin vector $f(r),g(r)\ge 0$0 is identical in the two phases. Third, the SWHV carries radial spin currents and exhibits a momentum-space distribution with two ring peaks at $f(r),g(r)\ge 0$1, whereas the HV is peaked at $f(r),g(r)\ge 0$2 (Chen et al., 4 Sep 2025). This suggests that the experimentally decisive distinction between the two states lies in momentum-space and interferometric observables rather than in static real-space spin density alone.

6. Stability and phase-diagram placement

The stability analysis proceeds by expanding

$f(r),g(r)\ge 0$3

and solving the Bogoliubov-de Gennes equations. For sufficiently large $f(r),g(r)\ge 0$4 and $f(r),g(r)\ge 0$5, the lowest non-zero modes remain gapped above zero, which identifies the SWHV as dynamically stable (Chen et al., 4 Sep 2025).

In the phase diagram reported by Chen et al., the SWHV region grows with increasing $f(r),g(r)\ge 0$6 and survives moderate interactions $f(r),g(r)\ge 0$7. Within the broader taxonomy of the paper, the SWHV belongs to the meron sector, in contrast to the DP and SS phases, which are combinations of meron and antimeron and are therefore termed meroniums. This situates the SWHV as the finite-momentum representative of a half-vortex meron phase in the trapped Rashba-coupled condensate.

7. Experimental identification

The summary specifies four experimental signatures for the SWHV (Chen et al., 4 Sep 2025).

First, in-situ spin-selective imaging of $f(r),g(r)\ge 0$8 and $f(r),g(r)\ge 0$9 reveals the same meron-like hole in one component and peak in the other as in the HV. Because this pattern is shared with the ordinary half vortex, it is informative about the meron structure but not by itself sufficient to isolate the spherical-wave character.

Second, time-of-flight or Bragg spectroscopy measures the momentum distribution. The SWHV exhibits concentric rings at $\int d^2r\,[f^2+g^2]=1,$0, whereas the HV is peaked at $\int d^2r\,[f^2+g^2]=1,$1. This is the clearest direct signature of the nonzero spherical-wave momentum.

Third, interferometry against a reference condensate reconstructs the local phase structure. The factor $\int d^2r\,[f^2+g^2]=1,$2 appears as radial interference fringes with spacing

$\int d^2r\,[f^2+g^2]=1,$3

Fourth, Stern-Gerlach separation combined with high-resolution imaging maps the Bloch vector $\int d^2r\,[f^2+g^2]=1,$4, thereby demonstrating the half-skyrmion, or meron, texture.

Taken together, these diagnostics establish the SWHV as a finite-momentum half vortex stabilized by Rashba spin-orbit coupling in a harmonic trap. Its defining characteristics are the condensate factor $\int d^2r\,[f^2+g^2]=1,$5, the meron spin texture with $\int d^2r\,[f^2+g^2]=1,$6, the interaction-driven transition from the $\int d^2r\,[f^2+g^2]=1,$7 HV phase, and dynamical stability against small perturbations in the parameter regime identified by the Bogoliubov-de Gennes analysis (Chen et al., 4 Sep 2025).