Topological defect

A topological defect is a localized obstruction to extending an ordered field smoothly and nonvanishingly through space. Its charge is determined by the homotopy class of the order parameter around the defect.

Disconnected vacuum components support walls, noncontractible vacuum loops support line defects, and nontrivial vacuum two-spheres support monopoles in three spatial dimensions. Allowed topology does not alone predict the number formed or their subsequent survival.

An extended order-parameter winding associated with a nontrivial third homotopy group of a vacuum manifold in three spatial dimensions. A texture can collapse and unwind when its field leaves the vacuum manifold; it differs from a permanent monopole or string core.

A three-dimensional point defect characterized by how the surrounding nematic director covers orientation space. Its hedgehog charge is not the same quantity as a planar disclination winding strength.

A point-defect texture that can compensate the radial boundary condition of a sphere with homeotropic anchoring, giving a dipolar particle–defect configuration in a uniform far field. It carries the opposite hedgehog charge to the radial particle boundary under a consistent convention.

A defect localized at a boundary of an ordered medium. A spherical inclusion with degenerate planar anchoring in a nematic liquid crystal can have a bipolar texture with two surface boojums, unlike a bulk point-defect pair.

A spontaneous symmetry breaking chooses order-parameter directions independently in regions that have not been in causal contact. Mismatches around disconnected components, loops or enclosing spheres can trap domain walls, cosmic strings or magnetic monopoles. The correlation length cannot exceed the causal horizon, so one monopole per horizon volume estimates the smallest causal initial abundance up to geometrical and efficiency factors.

// Target: classical-field-theory-soliton.bigb

A domain wall is a codimension-one configuration interpolating between disconnected components of a vacuum manifold. In a one-dimensional system with broken discrete symmetry it is a point defect of finite energy, so positional entropy produces a nonzero density at every positive temperature.

If the typical wall separation grows as cosmic time, surface energy per volume is of order wall tension divided by time. This decreases more slowly than the usual radiation-era total density and can lead to a wall-domination problem.

For a Landau-Ginzburg theory with positive gradient stiffness $\kappa$, $r<0$ and quartic coupling $u>0$, the degenerate minima are $\pm M_0$, $M_0^2=|r|/u$. A planar domain wall solves $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$. Multiplying by ${\varphi }^{\prime }$ and using the limiting minima gives $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$. Separating this equation yields the displayed profile, and integrating its excess energy gives the positive surface tension $2\sqrt{2\kappa }|r{|}^{3/2}/(3u)$. Its area cost is subextensive compared with bulk volume, permitting phase-separated common-tangent construction states when the average order parameter is constrained.

A vortex is a topological defect around which the phase of a complex order parameter has nonzero integer winding. In two dimensions an isolated vortex in a phase-only model has energy proportional to the logarithm of the system size.

The integer $q$ is a winding number: a counterclockwise circuit returns the same complex order parameter after its angle changes by $2\pi q$. Smooth changes that keep the order parameter nonzero on that circuit cannot change $q$. In a continuum superfluid, the core suppresses the magnitude, allowing the winding obstruction to end there; in an XY model the microscopic core instead lies beyond the smooth-field approximation. The vortex-antivortex binding figure illustrates a locally nonzero charge with zero charge on a surrounding large circuit.

An XY model orientation takes values on a circle, so a circuit around an excluded defect core can carry integer winding number. Smooth single-valued spin waves cannot change that integer without a core event or boundary crossing. The locally quadratic order-parameter phase action therefore describes a sector, rather than automatically summing every compact-field configuration. Vortex-antivortex unbinding changes the large-distance correlation function and renormalized phase stiffness, even though ordinary long-range order is already absent in two dimensions at positive temperature.

A quantum vortex has integer winding number of a condensate's complex argument. For atomic mass $m$, its superfluid velocity gives circulation $2\pi \hslash q/m$ for integer $q$. A straight vortex has a density-depleted core and, outside it, tangential velocity $q\hslash /(mr)$. Its long-range flow energy is proportional to $q^2\log (R/a)$, while its angular momentum has the sign of $q$.

Rotating a trap anisotropy can nucleate vortices in an atomic condensate. Expansion followed by absorption imaging resolves depleted cores, while varying rotation and waiting after removing the stirrer tests their preparation and persistence. The interpretation is quantized rotational response, rather than only the appearance of a condensed density profile.

In healing-length units a straight quantum vortex obeys $f^{\prime \prime }+f^{\prime }/r-{\mathcal{N}}^{2}f/r^2+(1-f^2)f=0$ and $f\to 1$. Inserting $f=1-a/r^2+\cdots$ makes its order-$r^{-2}$ balance $-{\mathcal{N}}^2+2a=0$, proving the tail. The physical healing length convention must be stated; the deficit of density $1-f^2$ has twice the leading coefficient of the amplitude deficit $1-f$.

For a centered quantum vortex in a two-dimensional Thomas–Fermi condensate, use an intermediate radius ${\xi }_{\mathrm{core}}\ll L\ll R$. The approximately uniform core contributes $\pi n_c{\hslash }^2{\mathcal{N}}^2[\log (L/\xi )+L_{0|\mathcal{N}|}]/m$. The outer superfluid velocity is $\hslash \mathcal{N}/(mr)$, so its kinetic contribution is $(\pi {\hslash }^2{\mathcal{N}}^2/m){\int }_L^R{n}_c(1-r^2/R^2)dr/r$. Integration cancels $L$ and produces the constant $-1/2$. The displayed result is the matched small-core approximation at fixed particle number, not a formula for arbitrarily large winding at fixed cloud size. The angular momentum is exactly $L_z=N\hslash \mathcal{N}$ for a normalized axisymmetric single-winding state.

For a cubic-quintic Gross–Pitaevskii equation with normalized bulk amplitude one, a straight quantum vortex of integer winding number $\mathcal{N}$ has radial equation $R^{\prime \prime }+R^{\prime }/r-{\mathcal{N}}^{2}R/r^2+(1-\alpha R^2-\beta R^4)R=0$. With $\alpha +\beta =1$, the derivative of the nonlinear term at $R=1$ is $-2(1+\beta )$. Balancing the order-$r^{-2}$ centrifugal term with the bulk amplitude restoring term proves the displayed tail for a stable nondegenerate bulk state, $1+\beta >0$. A vortex's amplitude tends to one; its full complex field tends to $e^{i\mathcal{N}\theta }$, not a direction-independent constant. At $\beta =-1$ the stated inverse-square expansion cannot cancel the centrifugal term.

If a continuous nonzero complex field tends uniformly to one on large circles, its values on a sufficiently large circle lie in a disk about one that excludes zero. That disk is contractible, so the field's winding number on that circle is zero. A lone unit-charge quantum vortex therefore cannot satisfy the literal boundary $\psi \to 1$. For a vortex it is the wave amplitude, rather than the entire complex field, that can approach one; $f(r)e^{i\theta }$ gives a simple illustration of direction-dependent far-field complex argument.

A quantum vortex becomes energetically favourable in a rotating reference frame when $\Delta E-\Omega \Delta L_z<0$. For a singly quantized vortex in a long radially Thomas–Fermi condensate, the cutoff-shell estimate isIntegrating the trapped number density against $u_{\theta }^2$ produces the logarithm and constant; integrating it against angular momentum per particle gives $\Delta L_z\simeq \pi H{n}_0\hslash R_r^2/2$. Resolved-core corrections can alter the nonlogarithmic constant. This criterion compares equilibrium rotating-frame energies and does not remove a dynamical vortex-entry barrier.

Single-valuedness of the condensate field gives total complex argument change $2\pi q$ on a closed loop avoiding zeros, where $q$ is an integer winding number. Integrating the superfluid velocity around that loop gives the displayed circulation. It is unchanged by smooth deformations of the loop that do not cross a quantum vortex.

If a condensate order parameter stays nonzero along a closed loop, its phase changes by an integer multiple of $2\pi$. Broken rotational symmetry does not destroy this single-valuedness constraint. In an inhomogeneous ring the winding integer need not label an angular-momentum eigenstate.

An event changing the phase winding of a condensate around a loop. A continuous nonzero order parameter cannot change its winding integer; a phase-slip path must pass through a locally vanishing amplitude or an equivalent vortex-crossing process. This use concerns a superfluid field, rather than only the relative phase of two classical oscillators.