Topological interface physics of defects and textures in spinor Bose-Einstein condensates

We provide a detailed description of our previously proposed scheme for topological interface engineering with constructed defects and textures perforating across coherent interfaces between different broken symmetries [Borgh and Ruostekoski, Phys. Rev. Lett. 109, 015302 (2012)]. We consider a spin-1 Bose-Einstein condensate, in which polar and ferromagnetic phases are prepared in spatially separated regions. We show that a stable coherent interface is established between the two phases, allowing defects of different topology to connect continuously across the boundary. We provide analytic constructions of interface-crossing defect solutions that could be experimentally phase imprinted using existing technology. By numerically minimizing the energy, we calculate the core structures of interface-crossing defect configurations. We demonstrate nontrivial core deformations to considerably more complex structures, such as the formation of an arch-shaped half-quantum line defect, an Alice arch, at the interface, with the topological charge of a point defect, whose emergence may be understood by the “hairy ball” theorem. Another example of an energetically stable object is the connection of a coreless vortex to a pair of half-quantum vortices. We show that rotation leads to spontaneous nucleation of defects in which a coreless vortex continuously transforms to a half-quantum vortex across the interface.

Figure 1

Nontrivial vortices in the FM phase. (a) A radial disgyration of the spin vector (red cones) around the singular core represents a spin vortex. This vortex carries a spin current but no mass circulation, a manifestation of the nonquantization of circulation alone in the FM phase. (b) All vortices in the class of singly quantized, singular vortices can be deformed into each other by local spin rotations. For example, it is possible to form a spin vortex with a cross-disgyration of the spin vector. (c) In addition to the one class of singular vortices, the FM phase also supports nontrivial, nonsingular vortices. In a coreless vortex the order parameter remains nonsingular everywhere in the vortex structure. The characteristic fountain structure of the spin is formed by a rotation of the spin vector around the vortex line, together with a winding of the condensate phase, corresponding to spin rotations about the local spin vector [indicated by the orthogonal green and blue (light and dark gray) vectors]. (d) Like the class of singular line defects, members of the family of nonsingular vortices are related by local deformations of the order parameter allowing different vortex configurations.

Figure 2

An analog of the field-theoretical Dirac monopole can be constructed in the FM phase of the spin-1 BEC. The radial hedgehog monopole structure of the spin vector (cones) appears as the termination of a doubly quantized, singular vortex line.

Figure 3

The polar phase of the spin-1 BEC supports half-quantum vortices, constructed as a $\pi$ winding of the condensate phase (which determines the quantized circulation) together with a $\widehat{\mathbf{d}}\to -\widehat{\mathbf{d}}$ winding of the nematic axis. The identification ${\zeta }^{\mathrm{p}}(\varphi ,\widehat{\mathbf{d}})={\zeta }^{\mathrm{p}}(\varphi +\pi ,-\widehat{\mathbf{d}})$ keeps the order parameter single valued. (a) Half-quantum vortex as described by Eq. (21). The nematic axis (cones) winds by $\pi$ in the plane perpendicular to the vortex line as the core is encircled. The disclination plane where the $\widehat{\mathbf{d}}\leftrightarrow -\widehat{\mathbf{d}}$ identification is made is indicated by the dashed line. (b) The winding of the nematic axis need not stay in the plane perpendicular to the vortex line. For example, a half-quantum vortex can also be constructed where $\widehat{\mathbf{d}}$ winds in a plane parallel to the vortex line as the defect is encircled.

Figure 4

Schematic illustrations of possible vortex connections. The polar phase is above the interface and the FM phase is below. (a) The simplest vortex connection to construct is to consider a singly quantized vortex in either phase. Even though these are different objects, representing different topologies, their wave functions can be joined continuously across the interface. (b) If the singly quantized vortex in the polar phase also contains a $2\pi$ winding of the nematic axis, its spinor wave function can be made to connect to a spinor representing a coreless vortex in the FM phase. (c) The FM Dirac monopole has a spinor structure similar to the coreless vortex (in fact, the monopole can be continuously unwound into a coreless vortex [56]) and it is therefore possible for it to form the termination point of a singly quantized polar vortex that ends at the interface. (d) Similarly, a polar monopole on the interface can form the termination point of a singly quantized FM spin vortex. The spinor is constructed by noticing that both defects have common phase windings of the spinor components.

Figure 5

Schematic illustrations of more complicated vortex connections that can be constructed from the basic defect solutions illustrated in Fig. 4. (a) A singly quantized vortex in the polar phase connecting to a doubly quantized vortex on the FM side may be cut in half at the interface and the resulting vortices in the two regions may be moved apart if an additional dark soliton plane is introduced in ${\zeta }_0$. (b) A singly quantized polar vortex can split into two half-quantum vortices when its energy relaxes [63]. A repulsive force between the half-quantum vortices makes the splitting energetically favorable. The splitting mechanism can then yield a state where a FM vortex connects to a pair of half-quantum vortices. (c) A Dirac dipole can be constructed by joining the Dirac strings of a Dirac monopole and an antimonopole [56]. Placed on the interface, the dipole connects to two singly quantized vortices on the polar side. (d) Rotation may cause interface-crossing vortex structures to nucleate. Here four nucleated vortex complexes are drawn together with a singular vortex already present.

Figure 6

Deformation of connecting singly quantized vortices. The green (gray) $|\langle \widehat{\mathbf{F}}\rangle |$ isosurface shows the interface and the cores of singular defects. The polar phase is above the interface. Energy relaxation causes the singly quantized vortex on the polar side to split into a pair of half-quantum vortices with $|\langle \widehat{\mathbf{F}}\rangle |>0$ cores on the polar side, recognizable by the characteristic $\pi$ winding of $\widehat{\mathbf{d}}$ [red (dark gray) cones]. These connect across the interface to a singly quantized FM vortex [cf. Fig. 5b]. A local rotation of the spin vector (light gray cones) allows the core region to fill with atoms with $|\langle \widehat{\mathbf{F}}\rangle |<1$. ($\left|c_2\right|=1.0\times {10}^4\hslash \omega l^3$ and $\Omega =0.20\omega$.)

Figure 7

Minimizing the energy of Eq. (24), corresponding to Fig. 4b results in a splitting of the singly quantized polar vortex, while the characteristic fountainlike spin structure of the coreless vortex is established in the FM part of the cloud. (a) The magnitude of the spin [$|\langle \widehat{\mathbf{F}}\rangle |=1$ is dark red (dark gray) with long arrows] shows the interface after relaxation of the energy, and the filled cores, with $|\langle \widehat{\mathbf{F}}\rangle |=1$ at the singularity, of two half-quantum vortices in the polar part. White arrows show the spin vector and indicate the coreless vortex in the FM part, and how the spin structure connects to the vortex cores across the interface. This result was obtained using $\left|c_2\right|=2.5\times {10}^2\hslash \omega l^3$ and $\Omega =0.12\omega$. (b) The half-quantum vortices may be identified by the winding of the nematic axis $\widehat{\mathbf{d}}$ (unoriented but shown as cones to emphasize winding). This displays the characteristic $\widehat{\mathbf{d}}\to -\widehat{\mathbf{d}}$ winding as any single vortex core is encircled. The two cores are joined by a disclination plane. Note that away from the core region the original topology of the singly quantized vortex is preserved. Here a stronger spin-dependent nonlinearity has been used to get more sharply defined FM cores: $\left|c_2\right|=1.0\times {10}^4\hslash \omega l^3$, $\Omega =0.19\omega$.

Figure 8

The condensate wave function varies smoothly across the polar-FM interface, showing that the defect states cross the interface smoothly, connecting the two topologies. (a)–(c) Densities in the individual spinor components ${\zeta }_{+}$,${\zeta }_0$ and ${\zeta }_{-}$, respectively. These vary smoothly and do not vanish simultaneously on the interface. The position of the interface, defined as the plane where $c_2$ changes sign, is indicated by the dashed line. (d) The smooth variation of the nonzero atomic density across the interface shows the continuity of the condensate wave function. (e) $\langle \widehat{\mathbf{F}}\rangle$ shown as color map [dark red (dark gray) with long arrows at $|\langle \widehat{\mathbf{F}}\rangle |=1$] and vector field (white arrows). Note how the spin structure connects smoothly across the interface, in particular connecting the FM cores of the half-quantum vortices continuously to the fountain structure of the spin in the coreless vortex. The magnitude of the spin also shows how the interface has acquired a finite width.

Figure 9

In Fig. 4d a polar point defect forms the termination of a FM spin vortex. As the energy relaxes, the point defect deforms into an archlike, singular defect line that connects to the interface at both ends. (a) An isosurface of the spin magnitude is shown in green (light gray). The spin magnitude rises to 1 (dark red, black) on the FM side of the interface ($z<0$) and inside the line defect. Gray cones indicate the spin vector. (b) Away from the defect, the $\widehat{\mathbf{d}}$ vector field (cones) in the polar phase retains the hedgehog structure of the original monopole. (c) In the trial wave function, Eq. (26), both the FM spin vortex and the polar monopole are formed by exactly overlapping vortex lines of opposite winding in ${\zeta }_{+}^{\mathrm{sv}\leftrightarrow \mathrm{pm}}$ and ${\zeta }_{-}^{\mathrm{sv}\leftrightarrow \mathrm{pm}}$. Constant-density surfaces for $n{\left|{\zeta }_{+}\right|}^2$ (red, medium gray) and $n{\left|{\zeta }_{-}\right|}^2$ (blue, dark gray) show how the arch-shaped line defect is formed by deformation of these vortex lines, such that they no longer overlap close to the interface. The half-quantum vortex line (above the interface) and the spin vortex (below the interface) are indicated by silver and gold (light gray) spin isosurfaces at $|\langle \widehat{\mathbf{F}}\rangle |=0.9$ and $|\langle \widehat{\mathbf{F}}\rangle |=0.5$, respectively. ($\left|c_2\right|=5.0\times {10}^2\hslash \omega l^3$ and $\Omega =0$.)

Figure 10

Deformation of the hedgehog point defect (left) on the interface into an arch-shaped line defect (right). Red (gray) cones represent the (unoriented) $\widehat{\mathbf{d}}$ field; the defect cores are shown as a green (gray) sphere and torus, respectively. The deformation creates a hole in the point-defect core, allowing it to expand into the line defect by locally deforming the $\widehat{\mathbf{d}}$ field. Away from the line defect, the asymptotic hedgehog structure is preserved. (Dashed lines indicate unperturbed $\widehat{\mathbf{d}}$ field for reference.) On any closed loop through the arch, $\widehat{\mathbf{d}}$ therefore winds to $-\widehat{\mathbf{d}}$, identifying the defect as a half-quantum vortex line. The line-defect core is filled with atoms with $|\langle \widehat{\mathbf{F}}\rangle |>0$; spin vector indicated by silver arrows.

Figure 11

Left: Isosurface of $|\langle \widehat{\mathbf{F}}\rangle |$ showing the cores of singular defects, together with $\widehat{\mathbf{d}}$ (red or dark gray cones) on the polar side of the interface, and $\langle \widehat{\mathbf{F}}\rangle$ (light gray cones) in the FM part. Four nucleated vortices are identified as polar half-quantum vortices connecting across the interface to nonsingular coreless vortices. Right: Top-down view of $\langle \widehat{\mathbf{F}}\rangle$ in the FM part, showing the four coreless vortices.