--- title: Spherical Wave Half Vortex (SWHV) url: https://www.emergentmind.com/topics/spherical-wave-half-vortex-swhv type: topic --- # Spherical Wave Half Vortex (SWHV) 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 [2509.03849]. ## 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 \(a_\perp^{-1}\) [2509.03849]. 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, \(\psi_\uparrow\) carries no angular phase, whereas \(\psi_\downarrow\) carries a \(+1\) vortex [2509.03849]. The local Bloch vector is \[ \mathbf{n}(r,\theta) = \frac{\Psi^\dagger\boldsymbol{\sigma}\Psi}{\Psi^\dagger\Psi} = \bigl(\sin\alpha(r)\cos\beta(\theta),\,\sin\alpha(r)\sin\beta(\theta),\,\cos\alpha(r)\bigr), \] with \[ \cos\alpha(r)=\frac{f^2(r)-g^2(r)}{f^2(r)+g^2(r)}, \qquad \beta(\theta)=\theta. \] Within this parametrization, the SWHV is mapped to a meron texture. At \(r\to 0\), \(g(0)\approx 0\), implying \(\mathbf n\approx(0,0,+1)\). At \(r\to\infty\), \(f\approx g\), so \(\alpha\to \tfrac{\pi}{2}\) and \(\beta=\theta\); the spins then lie in-plane and wind once during a full \(2\pi\) circuit in \(\theta\), 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 [2509.03849]. ## 3. Half-integer topology The topological characterization is given by the skyrmion-type charge \[ Q = \frac1{4\pi}\int d^2r\; \mathbf{n}\cdot \bigl(\partial_x\mathbf{n}\times\partial_y\mathbf{n}\bigr), \] for which the SWHV has \[ Q=+\tfrac12. \] The half-integer value is obtained by substituting the above \(\mathbf n(r,\theta)\) into the topological density in polar coordinates [2509.03849]. 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 \(e^{i k_c r}\) 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 \(\hbar\omega=1\) and \(a_\perp=\sqrt{1/m\omega}\), the mean-field energy of the \(j_z=\tfrac12\) 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\\[-3pt] &\quad +\tfrac12m\omega^2r^2(f^2+g^2) +\tfrac{N}{2}\bigl[(g+g')\,(f^2+g^2)^2 + (g-g')\,(f^2-g^2)^2\bigr] \Bigr\}, \end{align} where \(k_0\) is the Rashba-coupling parameter and \(g,g'\) are the intra-/inter-species couplings. The ground state is obtained by numerical minimization over \(\{f,g,k\}\) [2509.03849]. The variational outcome separates the HV and SWHV regimes. For \(g'g\), the minimum shifts to \(k_c=0\), giving the ordinary HV. The phase structure in \((g'/g,k_0)\) space contains a first-order transition at a critical line \(g'_c(g,k_0)\). This establishes the SWHV not as an excited-state deformation of the HV, but as a distinct variational minimum selected by the interplay of Rashba coupling, trap confinement, and interaction anisotropy. ## 5. Relation to the ordinary half vortex The ordinary half vortex is \[ \Psi_{\rm HV}(r,\theta)= \begin{pmatrix} f(r)\\ g(r)e^{i\theta} \end{pmatrix}, \] whereas the SWHV is \[ \Psi_{\rm SWHV}(r,\theta)= e^{i k_c r} \begin{pmatrix} f(r)\\ g(r)e^{i\theta} \end{pmatrix}. \] The difference is therefore entirely encoded in the radial phase factor [2509.03849]. | Property | HV | SWHV | |---|---|---| | Wavefunction | \(\bigl(f(r),\,g(r)e^{i\theta}\bigr)^T\) | \(e^{i k_c r}\bigl(f(r),\,g(r)e^{i\theta}\bigr)^T\) | | In-situ densities | \(n_{\uparrow,\downarrow}=f^2,g^2\) | Density unchanged | | Momentum-space structure | Peak at \(k=0\) | Two ring peaks at \(|\mathbf k|\approx k_c\) | The source summary emphasizes three consequences of this comparison. First, HV and SWHV have identical in-situ densities \(n_\uparrow=f^2\) and \(n_\downarrow=g^2\). Second, the local spin vector \(\mathbf n(r,\theta)\) is identical in the two phases. Third, the SWHV carries radial spin currents and exhibits a momentum-space distribution with two ring peaks at \(|\mathbf k|\approx k_c\), whereas the HV is peaked at \(k=0\) [2509.03849]. 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 \[ \Psi=\Psi_{\rm SWHV}+\delta\Psi \] and solving the Bogoliubov-de Gennes equations. For sufficiently large \(k_0 a_\perp\gtrsim 0.3\) and \(g'