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An antikink is a scalar-field kink with the opposite orientation of its vacuum boundary values. If ${\varphi }_K(x)$ joins vacua ${\varphi }_{-}$ to ${\varphi }_{+}$, then ${\varphi }_K(-x)$ joins ${\varphi }_{+}$ to ${\varphi }_{-}$. For an even double-well potential and an odd centered phi-four kink, the centered antikink is $-{\varphi }_K(x)$. With $Q=[\varphi (+\infty )-\varphi (-\infty )]/({\varphi }_{+}-{\varphi }_{-})$, a kink has $Q=1$ and an antikink has $Q=-1$. Their localized energies coincide by spatial reflection.

A spatially homogeneous field value minimizing the potential energy. For a finite-energy field configuration on an infinite domain, the potential is normally shifted so the relevant vacuum energy density is zero. Disconnected components of the vacuum manifold allow a scalar-field kink to connect distinct vacuum values.

The topological degree assigns an integer to a continuous map $f:M\to N$ between closed, connected, oriented manifolds of the same dimension $d$. Their top homology groups are infinite cyclic, generated by their fundamental classes. The induced map definesFor a smooth map and a regular value $y$, its equivalent local description iswhere the sign records preservation or reversal of orientation. The preimage is a finite set because the manifolds are compact and $y$ is regular. This is a signed count; opposite-orientation preimages cancel. If $\omega$ is a normalized target volume form with ${\int }_N\omega =1$, the pullback of a differential form gives the equivalent integral $\mathrm{deg}f={\int }_M{f}^{\ast }\omega$.

The homotopy invariance of mapping degree follows because homotopic maps induce the same homology map. In the smooth description, apply Stokes theorem to a homotopy $F:M\times [0,1]\to N$:since a top-dimensional form on $N$ is closed. Thus degree cannot change during a smooth deformation preserving the relevant boundary conditions. Degrees multiply under composition, and a nonzero-degree map is surjective: a missed target point would give a regular value with an empty preimage and hence degree zero.

For maps $S^d\to S^d$, the homotopy group ${\pi }_d(S^d)=\mathbb{Z}$ is completely detected by degree. This supplies integer topological sectors for many classical field-theory solitons. It is important to distinguish two ways spheres arise. In a nonlinear sigma model with the field approaching one fixed target value at infinity, compactify all of ${\mathbb{R}}^d$ to $S^d$ and classify the resulting map into the target. For a defect whose vacuum field varies with direction, retain the sphere $S_{\infty }^{d-1}$ surrounding the core and map it into the vacuum manifold. This is the vacuum-boundary degree as a defect charge. An asymptotic limit is a boundary hypothesis; finite energy alone in every possible model need not imply one.

For an Abelian Higgs vortex, the phase at infinity defines $S^1\to U(1)\cong S^1$. Its degree is the winding numberNonzero $n$ forces a zero of the Higgs field inside: a nowhere-vanishing field would extend its normalized phase through the disk and make the boundary loop contractible. This is also the flux classification of the Bogomolny vortex equations.

For a 't Hooft-Polyakov monopole, the normalized adjoint Higgs field $\hat{\Phi }$ gives $S_{\infty }^2\to SU(2)/U(1)\cong S^2$. With outward orientation its degree isThe integrand is the normalized target area form, so the hedgehog ansatz for a monopole has degree one. Nonzero degree obstructs extending the normalized Higgs direction through the ball; the full Higgs field must leave the vacuum manifold, normally vanishing in its core. The asymptotic unbroken Abelian magnetic flux measures the same integer with a convention-dependent sign.

For the Skyrme model, a field $U:{\mathbb{R}}^3\to SU(2)$ with $U\to 1$ at infinity gives $S^3\to S^3$, because SU(2) as the three-sphere identifies the target. A choice of orientations gives its topological baryon number in the Skyrme model as the winding integral $\pm (24{\pi }^2{)}^{-1}\int \mathrm{tr}(U^{-1}dU{)}^3$; specifying the orientation fixes the sign. Similarly, the transition map at infinity of a four-dimensional Yang-Mills instanton has degree in ${\pi }_3(SU(2))$, measured by its Chern charge.

A scalar-field kink illustrates the discrete-vacuum case: its endpoint data lie in ${\pi }_0(\{\pm 1\})$, rather than being a map between connected positive-dimensional spheres. For the double-well field its topological charge is $Q=[\varphi (+\infty )-\varphi (-\infty )]/2$. A kink has $Q=1$, an antikink $Q=-1$; they cannot individually deform to vacuum while their endpoints are fixed.

Degree classifies topology, rather than every solution. Smooth motion or local perturbations with fixed boundary data preserve it, and a nonzero degree obstructs decay to the vacuum within that class. It does not guarantee existence of a smooth energy minimizer, a fixed size, uniqueness, or dynamical stability. The degree and energetic stability of a field configuration are distinct: a field can shrink towards a singular limit at fixed degree, as Derrick's theorem diagnoses in many two-derivative theories. Moreover degree need not classify all homotopy classes for an arbitrary target; other homotopy groups or invariants may be needed. Within one degree there can be continuous moduli spaces and several stationary solutions.

In this static scalar problem, a classical field-theory soliton is a nontrivial smooth localized finite-energy field configuration which is static in its rest frame and stable against admissible small perturbations. Moving examples are obtained by a Lorentz boost. The relevant static energy isUse the usual vacuum normalization $U\ge 0$, with zero energy at a vacuum, and impose finite-energy vacuum behavior at spatial infinity. These assumptions, together with the unconstrained real target and the canonical two-derivative kinetic term, are the hypotheses of the following Derrick theorem argument.

For Derrick scaling ${\varphi }_{\lambda }(x)=\varphi (\lambda x)$, change variables to obtainA static solution must be stationary under this admissible scale variation. The Derrick virial identity isFor $D>2$, both terms are nonpositive and the gradient term is strictly negative for a nonconstant field, so no nontrivial static solution is possible. At $D=2$, the identity forces $V=0$, but the gradient term is scale invariant and a further argument is needed.

Since $U\ge 0$, $V=0$ implies $U(\varphi (x))=0$ everywhere. Every zero of a differentiable nonnegative potential has $U^{\prime }=0$. The static Euler-Lagrange field equation $\Delta \varphi =U^{\prime }(\varphi )$ therefore makes $\varphi$ a harmonic function. Each derivative ${\partial }_i\varphi$ is harmonic and belongs to L2 space by finite energy. Its mean-value property and the Cauchy-Schwarz inequality on a ball of radius $R$ implyLetting $R\to \infty$ gives ${\partial }_i\varphi =0$. This proves the two-dimensional flat-target Derrick obstruction. Thus a nontrivial static real-scalar soliton in this class can occur only for $D=1$. The argument does not apply unchanged to curved-target sigma models or gauge-field energies.

In one dimension the scale identity is $T=V$, and the second scale derivative is $2V>0$ for a nontrivial solution; scale variation does not rule it out. To derive the Bogomolny equations, write $U=\frac{1}{2}{W}^{\prime }(\varphi {)}^2$ on the field interval of interest. Completing the square givesTherefore the Bogomolny bound and its saturation equation areEquivalently, for a monotone field connecting adjacent vacua, ${\varphi }^{\prime }=\pm \sqrt{2U(\varphi )}$, with the sign chosen for its direction. Differentiating the first-order equation gives ${\varphi }^{\prime \prime }=W^{\prime }{W}^{\prime \prime }=U^{\prime }$, verifying the static Euler-Lagrange field equation. Saturation minimizes energy within the fixed-boundary topological sector.

For the given sextic potential, factorization gives $U(\varphi )={\varphi }^2({\varphi }^2-4{)}^2$, so the vacua are $-2,0,2$. ChooseThis choice satisfies $U=W^{\prime 2}/2$. On $0<\varphi <2$, the increasing kink in a phi-six model satisfies ${\varphi }^{\prime }=\sqrt{2}\varphi (4-{\varphi }^2)$. Setting $y={\varphi }^2$ gives $y^{\prime }=2\sqrt{2}y(4-y)$. Separation givesand henceThe endpoints are zero and two, and the mass follows from $W(2)-W(0)=4\sqrt{2}$. The other increasing elementary scalar-field kink isIt uses the opposite sign of $W^{\prime }$ and has the same mass. Reflecting $x-x_0$ yields the two decreasing antikinks; $x_0$ is the free center.

There is no single finite-width static kink directly connecting $-2$ to $2$. Multiplying the static equation by ${\varphi }^{\prime }$ gives the first integral ${\varphi }^{\prime 2}/2-U=0$ for a finite-energy vacuum-to-vacuum solution. At the intermediate vacuum $\varphi =0$, this forces ${\varphi }^{\prime }=0$. Uniqueness for the smooth second-order field equation makes data $(\varphi ,{\varphi }^{\prime })=(0,0)$ stay at that vacuum, so a nontrivial solution cannot cross it at finite $x$. This is the intermediate-vacuum obstruction to a kink; the two elementary transitions can only separate by an infinite interval.

WriteThe four vacua are $-q,-p,p,q$. Hence there are three adjacent scalar-field kink sectors, $(-q,-p)$, $(-p,p)$ and $(p,q)$, together with their three reversed antikinks. Withthe Bogomolny bound gives $E=|W({\varphi }_{+})-W({\varphi }_{-})|$. The central kink obeys ${\varphi }^{\prime }=W^{\prime }(\varphi )$ and hasThe two outer kinks obey ${\varphi }^{\prime }=-W^{\prime }(\varphi )$ and have equal energyFor the kink passing through zero, choose $\varphi (0)=0$. Its profile is determined implicitly byIf $q=p$, the central equation becomes ${\varphi }^{\prime }=(p^2-{\varphi }^2{)}^2$, soFor $q=p+\epsilon$,so the leading ratio is

Differentiating the potential givesThus $\varphi =\pm 1$ are stationary points, andbecause $|ϵ|<1$. Both are local minima. Their energies are

For $ϵ=0$, the static energy can be completed to a square:The increasing scalar-field kink therefore obeys the first-order Bogomolny equationWith center $X$, its solution and energy areThe antikink uses the opposite sign.

For small positive $ϵ$, the true vacuum $\varphi =-1$ lies below the false vacuum $\varphi =1$ byThis pressure exerts force $\Delta U$ on a kink with $-1$ on its left and $+1$ on its right. Dividing by its leading mass $4/3$ gives acceleration toward the false-vacuum side:

An antikink followed by a kink encloses a region of the lower vacuum while approaching $\varphi =1$ at both infinities. Vacuum pressure pushes the pair apart, whereas their attraction pulls them together. At a static separation $s$,soThis estimate is self-consistent for $ϵ\ll 1$, when the two soliton cores are well separated.

Write the scalar field energy as $E=T+V$, whereUnder the Derrick scaling ${\varphi }_{\lambda }(x)=\varphi (\lambda x)$, a change of variables givesA finite-energy solution of the Euler-Lagrange equation is stationary under this admissible variation. The Derrick virial identity is therefore

The static field equation is ${\varphi }^{\prime \prime }=U^{\prime }(\varphi )$, so integration givesThe polynomial before $C$ is nonnegative and vanishes, so requiring the minimum to be zero fixes $C=0$. Hence the vacuum manifold iswhich has three elements. A finite-energy scalar-field kink can join only adjacent vacua: a solution cannot cross the intermediate vacuum at finite $x$ because its first integral would have $\varphi ={\varphi }^{\prime }=0$ there and the Picard-Lindelöf theorem would make it constant. There are therefore four oriented topological sectors,comprising two increasing kinks and their two antikinks. Symmetry under $\varphi \mapsto -\varphi$ and spatial reflection generates all four from one profile.

For the $0\to 1$ sector, completing the square gives the Bogomolny boundEquality holds for the Bogomolny equationWith $y={\varphi }^2$, this becomes the logistic differential equation $y^{\prime }=2y(1-y)$. Translation invariance supplies an arbitrary center $x_0$, and the explicit kink in a phi-six model isIt tends to $0$ and $1$ at the two spatial ends and saturates the bound. Its mass is consequently

A scalar-field vacuum is a constant field at a minimum of its potential energy, with the minimum shifted to zero when discussing finite-energy classical field-theory solitons. Distinct isolated vacua allow a scalar-field kink to approach different values at the two ends of space.