Why is it interesting?

The scalar model in 1 spatial and 1 time dimension is the simplest quantum field theory we can study.

It is often used to introduce the basics of solitonic solutions, since the Scalar 1+1 model contains a non-trivial kink solution.

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The model only contains a real scalar field $\phi (x) \in \mathbb{R}$. The scalar potential is

$$ U= \frac{\lambda}{2} ( \phi^2-a^2)^2, $$

and is conveniently normalized such that the ground state (the vacuum solution) is at $U=0$. In addition, the potential is bounded from below $U\geq 0$.

The field equation is

$$ \partial_{x_i}^2 \phi = 2 \lambda \phi (\phi^2-a^2). $$

It is important to note that this field equation is a non-linear differential equation and hence allows non-linear solutions.

The field equation can be either derived from the action of the model or from the energy functional:

$$ E= \int dx \left[ \frac{1}{2} \left( \frac{\partial \phi}{\partial x}\right)^2 + U(\phi)\right]. $$

Then, demanding that the variation $\delta E$ vanishes yields the field equation.

Standard Solutions

The standard solutions of the field equation are simply

$$ \phi_c = \pm a. $$

These two solutions are the classical vacuum solution.

Solitonic Solutions - The Kink

In addition, we can find other less trivial solutions:

$$ \phi_k^\pm = \pm a \tanh(\sqrt{\lambda} ax_i) .$$

These solutions are called the kink and the anti-kink solution. It is important to note that these solutions are time-independent. In this sense they are non-dissipative, i.e. do not vanish over time like normal wave solutions do.

We can check that $\phi_k^\pm $ is a solution by putting it into the field equations.

Check that kink solutions solve the field equation

1. First, rescale the field: $\phi = a \tilde{\phi} .$ This simplifies the field equation to $ \partial_{x_i}^2 \tilde{\phi} = 2 \tilde{\lambda} \tilde{\phi} (\phi^2-1).$
2. Then use that $\tanh = \frac{\sinh}{\cosh}$, $\sinh^2-\cosh^2 = 1 $ and $\partial_x \tanh(x) = \frac{1}{\cosh(x)}$,

On the left-hand side of the field equation we get if we put $\phi_k^+$ in:

\begin{align} \partial_{x_i}^2 \tilde{\phi_k^+} &= \partial_{x_i}^2 \tanh(\sqrt{\tilde{\lambda}} x_i) \\ & = - 2 \tilde{\lambda} \frac{\sinh}{\cosh^3}.\end{align}

On the right-hand side, ignoring all constants and only focussing on the functions, we get

\begin{align} \tilde{\phi_k^+} ((\phi_k^+)^2-1) &\propto \tanh (\tanh^2-1) \\ &= \tanh \left(\frac{\sinh^2}{\cosh^2}-1\right) \\ &= \tanh \left(\frac{\sinh^2}{\cosh^2}-\frac{\cosh^2}{\cosh^2}\right) \\ &= \tanh \left(\frac{\sinh^2-\cosh^2}{\cosh^2}\right) \\ &= \tanh \frac{1}{\cosh^2} \\ &= \frac{\sinh}{\cosh} \frac{1}{\cosh^2} \\ &= \frac{\sinh}{\cosh^3} .\\ \end{align}

Thus, the right-hand side and the left-hand side of field equations yield exactly the same when we put $\phi_k^+$ in and therefore these functions really solve the field equations.

Properties of the Kink Solution

We are interested in finite energy solution since infinite energy solutions would require infinite energy to create.

To make sure that our kink solution has finite energy we need the boundary conditions that the scalar field only takes ground state values at infinity, i.e. either $a$ or $-a$. The ground state has zero energy and hence this requirement basically just means that we demand that the field energy vanishes at infinity.

We can write these boundary conditions mathematically:

$$ \phi(\infty) - \phi(-\infty) \stackrel{!}{=} 2na, $$ where $n \in (0,-1,+1)$.

These boundary conditions become important when we discover that there is a conserved current in the model:

$$ j_\mu (x) = \epsilon_{\mu \nu} \partial^\nu \phi(x), $$ where $\epsilon_{\mu \nu} $ is the two-dimensional Levi-Civita symbol. This current is conserved since partial derivatives commute and $\epsilon_{\mu \nu}$ is totally antisymmetric. A sum over something symmetric $\partial \mu \partial^\nu$ times something totally antisymmetric $\epsilon_{\mu \nu}$ yields zero:

$$ \partial^\mu j_\mu (x) = 0. $$

Whenever we have conserved current, we can find a conserved charge:

\begin{align}Q &= \int_{-\infty}^\infty dx j_0 \notag \\ & = \int_{-\infty}^\infty dx ( \epsilon_{0 0} \partial^0 \phi(x) + \epsilon_{0 1} \partial^1 \phi(x) ) \\ & = \int_{-\infty}^\infty dx \partial^1 \phi(x) \\ & = \int_{-\infty}^\infty dx \partial^x \phi(x) \\ &= \phi(\infty) - \phi(-\infty) = 2na \, . \end{align}

The conserved quantity is, therefore, $n$, up to a constant factor. This conserved quantity is known as winding number of the kink and because of this topological conserved quantity the kink is stable.

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Examples

Example1

Example2:

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