Kinks and Domain Walls

A kink is a finite-energy scalar configuration that interpolates between disconnected vacua along one spatial direction. In the $1+1$-dimensional $\phi^4$ model the interpolation is explicit, its tension saturates a first-order energy bound, and its fluctuation operator has a normalizable translation zero mode but no negative mode. The same transverse profile gives a domain wall in higher dimensions, with tension equal to energy per unit wall area.

Required background. Finite-energy boundary data identifies the kink sector, and scalar interactions and stability supplies the potential and small-fluctuation framework. Helpful background. Linear ODEs and Wronskians helps with the one-dimensional spectral problem.

The broken-symmetry scalar model

Take a real scalar in $1+1$ dimensions with the site’s mostly-minus metric,

$$\mathcal L =\frac12\partial_\mu\phi\,\partial^\mu\phi -V(\phi), \qquad V(\phi)=\frac{\lambda}{4}(\phi^2-v^2)^2 ,$$

where $\lambda,v>0$. The two vacua are $\phi=\pm v$, and the mass of a small fluctuation about either vacuum is

$$m^2=V''(\pm v)=2\lambda v^2 .$$

For a static field, finite energy requires $\phi(\pm\infty)=\pm v$ in the kink sector. The Euler–Lagrange equation is

$$\phi''=\lambda\phi(\phi^2-v^2).$$

Multiplying by $\phi'$ and integrating once gives

$$\frac12(\phi')^2-V(\phi)=C .$$

The vacuum boundary conditions set $C=0$. For the orientation $-v\to+v$,

$$\phi' =\sqrt{\frac{\lambda}{2}}\,(v^2-\phi^2).$$

Separation of variables then yields

$$\phi_{\mathrm K}(x;X) =v\tanh\!\left[\frac{m}{2}(x-X)\right].$$

The parameter $X$ is arbitrary because translations are an exact symmetry. Reversing the boundary orientation gives the antikink. This derivation and its mechanical analogy are treated in Coleman 1985, § 2.1, pp. 187–191 and Manton and Sutcliffe 2004, §§ 5.1–5.2, pp. 109–116.

Tension and the first-order bound

Choose a smooth function $W$ such that

$$(W'(\phi))^2=2V(\phi), \qquad W'(\phi):=\sqrt{\frac{\lambda}{2}}\,(v^2-\phi^2).$$

The sign is chosen for the orientation $-v\to+v$. On the interval $-v\leq\phi\leq v$ this is the positive square root of $2V$; the polynomial definition remains smooth outside that interval. The energy is

$$\begin{aligned} E &=\int\mathrm dx\, \left[ \frac12\big(\phi'-W'(\phi)\big)^2 +\phi'W'(\phi) \right]\\ &\geq W(v)-W(-v). \end{aligned}$$

With $W(\phi)=\sqrt{\lambda/2}\,(v^2\phi-\phi^3/3)$, the kink tension is

$$T_{\mathrm K} =\frac{2\sqrt{2\lambda}}{3}v^3 =\frac{m^3}{3\lambda}.$$

The first-order profile saturates the bound. Because the square is nonnegative, it is an absolute minimum within the fixed endpoint sector. This conclusion uses the boundary term and its orientation; changing the endpoints or the sign without changing the first-order equation is an error.

The virial identity supplies an independent check. In one spatial dimension, the kink obeys $\tfrac12(\phi')^2=V$ pointwise, hence $E_2=E_0=T_{\mathrm K}/2$, exactly as Derrick scaling requires.

Small fluctuations and the translation mode

Write

$$\phi(t,x)=\phi_{\mathrm K}(x;X)+\eta(x)e^{-i\omega t}$$

and linearize. The fluctuation equation is

$$\mathcal H_{\mathrm K}\eta=\omega^2\eta, \qquad \mathcal H_{\mathrm K} =-\frac{\mathrm d^2}{\mathrm dx^2} +m^2\left[ 1-\frac32\operatorname{sech}^2 \!\left(\frac{m(x-X)}{2}\right) \right].$$

The square completion also factorizes this operator. On the physical $L^2(\mathbb R)$ domain, with boundary terms vanishing, define

$$\mathcal A :=\frac{\mathrm d}{\mathrm dx}-W''(\phi_{\mathrm K}), \qquad \mathcal H_{\mathrm K}=\mathcal A^\dagger\mathcal A, \qquad \mathcal A\phi_{\mathrm K}'=0 .$$

Consequently $\langle\eta,\mathcal H_{\mathrm K}\eta\rangle=\lVert\mathcal A\eta\rVert^2\geq0$, so there is no negative eigenvalue. Differentiating the static field equation independently gives the same zero-mode equation, $\mathcal H_{\mathrm K}\phi_{\mathrm K}'=0$. The normalized translation mode is

$$\eta_0(x) :=\sqrt{\frac{3m}{8}}\, \operatorname{sech}^2\!\left(\frac{m(x-X)}{2}\right), \qquad \int_{-\infty}^{\infty}\mathrm dx\, \lvert\eta_0(x)\rvert^2=1 .$$

It moves the kink rather than destabilizing it. The same Pöschl–Teller operator has one additional bound “shape” mode with $\omega^2=3m^2/4$ and continuum states beginning at $\omega^2=m^2$. Coleman 1985, § 2.2, pp. 191–193 explains the general stability argument, while Manton and Sutcliffe 2004, § 5.2, pp. 111–116 give the profile and spectrum in this normalization after translating their coupling conventions.

An important distinction follows:

• the boundary sector prevents continuous unwinding;
• the square completion proves energetic minimality in that sector;
• the nonnegative self-adjoint Hessian excludes exponentially growing linear modes once the translation direction is treated as motion along the symmetry orbit;
• the normalizable zero mode supplies a collective coordinate;
• none of these statements computes the renormalized quantum kink mass or excludes every quantum process in a different theory.

From a kink to a domain wall

In $D$ spatial dimensions, let the field depend only on one transverse coordinate $z$. The same profile $\phi_{\mathrm K}(z-X)$ solves the field equation and has energy

$$E=T_{\mathrm W}\int\mathrm d^{D-1}y , \qquad T_{\mathrm W}=T_{\mathrm K},$$

where $T_{\mathrm W}$ is tension, not a finite total energy on an infinite wall. The defect has codimension one. Slowly varying its position to $X=X(t,\mathbf y)$ produces a worldvolume field. Its metric coefficient is the zero-mode norm,

$$G_{XX} :=\int_{-\infty}^{\infty}\mathrm dz\, \big(\partial_X\phi_{\mathrm K}\big)^2 =\int_{-\infty}^{\infty}\mathrm dz\, \big(\phi_{\mathrm K}'\big)^2 =T_{\mathrm K}.$$

The last equality follows because the first-order kink has energy density $(\phi_{\mathrm K}')^2$. To quadratic order in worldvolume derivatives,

$$S_{\mathrm{eff}} =-T_{\mathrm W}\int\mathrm d^D y +\frac{T_{\mathrm W}}{2} \int\mathrm d^D y\, \partial_\alpha X\,\partial^\alpha X+\cdots .$$

The derivative expansion requires wall curvature radii and wavelengths much larger than $m^{-1}$, the wall thickness. Large curvature, wall intersections, radiation into bulk modes, and cosmological network evolution are separate problems.

The shared boundary-family map locates the wall among defects classified by transverse infinity: its $S^0_\infty$ data are not the $S^1_\infty$ winding of a vortex or the $S^2_\infty$ data of a monopole. The soliton boundary and stability comparison records separately the kink’s explicit construction, virial relation, fluctuation index, position modulus, and quantum limitation.

Checks and limitations

Dimensions. In $1+1$ dimensions, $[\phi]=0$, $[v]=0$, $[\lambda]=2$, and $[T_{\mathrm K}]=1$, as required for a particle mass. In higher dimensions the same formula uses the dimensionful parameters of that theory and gives energy per wall area.

Tails. Linearizing near $\phi=v$ gives $\delta\phi\sim e^{-m|x|}$, consistent with the exact $\tanh$ tail. A finite numerical box must extend several $m^{-1}$ beyond the core.

Orientation. The sign of $Q_{\mathrm K}$ and of the first-order equation changes under $x\mapsto-x$. The positive tension does not.

Quantum scope. Loop corrections require a regulator, counterterms, vacuum subtraction, and treatment of the zero mode. They are not determined by the classical profile and are intentionally not derived here.

Common pitfalls

Calling tension a total energy for an infinite wall. A kink in one spatial dimension has finite energy. A planar wall in higher dimensions has finite energy per unit area and infinite total energy in infinite volume.

Removing the zero mode as a numerical error. Translation invariance requires it. A small nonzero eigenvalue in a finite box measures boundary and discretization effects; it should converge to zero as both are controlled.

Inferring the whole spectrum from topology. Topology fixes the sector, while the shape mode and continuum come from the Hessian. The absence of a negative mode follows here from the explicit energy bound and spectral problem.

Exercises

1. Substitute the kink profile into the first-order equation and integrate its energy density to verify $T_{\mathrm K}=m^3/(3\lambda)$.

Solution

For $u=m(x-X)/2$, $\phi'=vm\,\operatorname{sech}^2u/2$. The first-order relation gives equal gradient and potential contributions, so the energy density is $(\phi')^2$. Therefore

$$T_{\mathrm K} =\frac{v^2m^2}{4}\frac{2}{m} \int_{-\infty}^{\infty}\mathrm du\,\operatorname{sech}^4u =\frac{v^2m}{2}\frac43 =\frac{m^3}{3\lambda}.$$

2. Show directly that $\phi_{\mathrm K}'$ is an unnormalized zero mode without using the explicit $\tanh$ profile, and relate it to the normalized mode $\eta_0$.

Solution

Differentiate the static equation $-\phi_{\mathrm K}''+V'(\phi_{\mathrm K})=0$ with respect to $x$. This gives

$$\left[-\frac{\mathrm d^2}{\mathrm dx^2} +V''(\phi_{\mathrm K})\right]\phi_{\mathrm K}'=0,$$

which is precisely $\mathcal H_{\mathrm K}\phi_{\mathrm K}'=0$. Normalizability follows because $\phi_{\mathrm K}'$ decays exponentially. Since

$$\int\mathrm dx\,(\phi_{\mathrm K}')^2=T_{\mathrm K},$$

the unit-normalized mode used above is $\eta_0=\phi_{\mathrm K}'/\sqrt{T_{\mathrm K}}$.

3. Promote the kink center to $X=X(t)$ and derive the quadratic kinetic term. Why would the same construction fail for a nonnormalizable zero mode?

Solution

For $\phi(t,x)=\phi_{\mathrm K}(x-X(t))$,

$$\partial_t\phi=-\dot X\,\phi_{\mathrm K}',$$

so the microscopic kinetic energy becomes

$$\frac12\int\mathrm dx\,(\partial_t\phi)^2 =\frac{\dot X^2}{2} \int\mathrm dx\,(\phi_{\mathrm K}')^2 =\frac{T_{\mathrm K}}{2}\dot X^2.$$

Thus $G_{XX}=T_{\mathrm K}$. If the zero-mode norm diverged, the effective kinetic coefficient would be infinite and no normalized fluctuation state would generate a finite collective coordinate in the stated volume and boundary conditions.

Continue

Bogomolny Bounds and First-Order Equations abstracts the square completion. Moduli-Space Dynamics and Collective Quantization promotes $X$ to a low-energy quantum coordinate.

References

• Coleman, Sidney. Aspects of Symmetry: Selected Erice Lectures. Cambridge University Press, 1985, ch. 6, §§ 2.1–2.2, pp. 187–193. DOI.
• Manton, Nicholas, and Paul Sutcliffe. Topological Solitons. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004, ch. 5, pp. 109–130. Chapter DOI.