# $\phi^4$ theory kinks as fermions?

In 1+1 dimensions there is duality between models of fermions and bosons called bosonization (or fermionization). For instance the sine-Gordon theory $$\mathcal{L}= \frac{1}{2}\partial_\mu \phi \partial^\mu \phi + \frac{\alpha}{\beta^2}\cos \beta \phi$$ can also be described in terms of fermions as the massive Thirring model $$\mathcal{L}= \bar{\psi}(i\gamma^\mu-m)\psi -\frac{1}{2}g \left(\bar{\psi}\gamma^\mu\psi\right)\left(\bar{\psi}\gamma_\mu\psi\right)$$ where the particle created by $\psi$ can be understood as a kink of sine-Gordon, and the particle created by $\phi$ can be understood as a bound state of two fermions from the Thirring model.

Unlike sine-Gordon, $\phi^4$ $$\mathcal{L}= \frac{1}{2}\partial_\mu \phi \partial^\mu \phi + \frac{1}{2}m^2\phi^2 -\frac{1}{4}\lambda \phi^4$$ has only two vacua in the broken symmetry phase. I'm wondering whether here too we can write fermionic creation operators for the kinks, and rewrite the theory as a local field theory of the kink fields?

The reason I think we can is that we can do this for the quantum Ising model which has much in common with $\phi^4$. The Ising model is defined on a 1d spin chain, and the ground states in the broken symmetry phase are where the 3rd component of the spins are either all pointing up or all down.

The operators $\psi_1(i),\psi_2(i)$ are defined at each lattice point $i$ in terms of Pauli matrices as $$\psi_1(i) = i\sigma_2(i)\prod_{\rho=-\infty}^{i-1}\sigma_1(\rho)$$ $$\psi_2(i) = \sigma_3(i)\prod_{\rho=-\infty}^{i-1}\sigma_1(\rho)$$ The infinite product part acts to flip the 3rd component of spin to create a kink, and the Pauli matrix part gives it the usual fermionic anticommutation relations.

It turns out in the continuum limit $\psi_{1,2}$ act like two components of a free Majorana fermion. Can $\phi^4$ also be expressed in terms of a Majorana fermion? What are the relations for the fermion field of $\phi^4$ that are analogous to the relations for $\psi_{1,2}$ in terms of Pauli matrices?

• This looks like an important question. I am scratching my head over this. Could you give references to the operators on the lattice and Majorana fermions, particularly if this has some reference to bosonization. – Lawrence B. Crowell Jun 8 '16 at 14:49
• It's discussed in chapter 9 of Giuseppe Mussardo's book Statistical Field Theory – octonion Jun 8 '16 at 16:54
• A crucial point in the Sine-Gordon/Massive Thirring duality is that both the Sine-Gordon solitons and the massive fermions in Thirring Model have a $\mathbb Z$ charge. It is topological for the former and Nother type for the latter. In the case of $\phi^4$ kinks, the topological charge is $\mathbb Z_2$ so the first thing to look for is another theory with a $\mathbb Z_2$ charge. – Diracology Jun 8 '16 at 19:31
• I don't fully understand the implications yet, but the free Majorana theory does have a Z2 symmetry. You can represent spacetime by a complex number $z=x+it$ and consider new fields $\Psi,\bar{\Psi}$ that have $\psi_{1,2}$ as real/imaginary components. Then you can rewrite the action as $\int d^2 z \Psi \partial_\bar{z} \Psi +\bar{\Psi}_z \bar{\Psi}+i m \bar{\Psi}\Psi$. This has a symmetry under flipping the sign of $\bar{\Psi}$ and $m$. – octonion Jun 8 '16 at 21:24
• This is not necessarily the case if you are in two dimensions or larger. The charge is a measure of the degeneracy of the vacuum, which is not going to be just two points in dim > 1. – Lawrence B. Crowell Jun 8 '16 at 21:25