# $\phi^4$ 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. Jun 8, 2016 at 14:49
• It's discussed in chapter 9 of Giuseppe Mussardo's book Statistical Field Theory Jun 8, 2016 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. Jun 8, 2016 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 $m$. Jun 8, 2016 at 21:24
• @octonion I saw that it was old but I think it's an interesting question. The usual way to study the Ising model with bosonization is to introduce two decoupled Ising models and then combine the two Majoranas into a Dirac fermion, see Zuber & Itzykson, PRD 15 10 (1977). Another interesting paper is Shankar & Murthy, PRB 72 224414 (2005), which studies the massive Schwinger model. This model can be bosonized and the Ising CFT is in the phase diagram, but the authors stress that the Majoranas have no simple relation to the original Dirac fermions. Perhaps these references will interest you. Jan 12, 2020 at 0:29

Quantum fluctuations in the kink sectors of Sin-Gordon and the quartic interaction theory are described by reflectionless Pöschl-Teller-Operators, which form a SUSY-Chain with $$N$$ elements. The second quantization is then obtained by the spectral decompostion of those operators that are connected through annihilation and creation operators. Sin Gordon corresponds to $$N=1$$ and $$\Phi^4-$$Theory to $$N=2$$. Because of the similarities of the quantum structure of both systems, there should be the possibility to construct a fermion model which corresponds to the quartic theory for certain values of the coulping constant, too. An interesting point is, that the eigenfunctions of the creation operators for the Nth system have the same structure as the $$N$$ body partition function in the Ising-Model, calculated in mean field approximation.

Edit: In short form: If you expand around the kink the second order term is called the fluctuation operator

$$\frac{\delta^2S_E}{\delta\phi(x)\delta\phi(x')}|_{\phi_{kink}=\phi}=[-\partial^2_x+V''(\phi_{kink}(x))]\delta(x-x')$$

The operators for both systems are of the form

$$F=-\partial^2_x+N^2m^2-N(N+1)m^2sech^2(mx)$$. With spectra

$$spec(F)=\{\omega^2_n\}_{n=0,...,l-1}\cup\{\omega^2_l\}_{l=N}\cup\{k^2+N^2m^2\}_{k\in\mathbb{R} }$$ This is a chain of supersymmetric operators. Another motivation for SUSY is this: The kink forms an equipotential surface in field configuration space due to its translational invariance. That means there is a parameter, where under change of it the action remains constant. So the fluctuations vanish. From this fact it follows, that the zero mode of the operator $$F$$ has to be the derivative along that parameter of the classical solution, which is a monotonous function, so the derivative has no nods and therefore it has to be the ground state of a supersymetrical quantum system. If you know the ground state of such a system you can evaluate the superpotential together with the ladder operators and recover the fluctuation operator from $$F=A^\dagger_NA_N$$.

$$A^\dagger_N=-\partial_x+Nmtanh(mx)$$ $$A_N=\partial_x+Nmtanh(mx)$$

The Nth system has $$N-1$$ bound states which are all smooth and faster decaying torwards infinity than any polynomial.

Sin-Gordon (normalized):

$$\omega^2_0=0$$ $$\psi_0(x)=\sqrt{\frac{m}{2}}sech(mx)$$

$$\phi^4$$ (normalized):

$$\omega^2_0=0$$ $$\psi_0(x)=\sqrt{\frac{3m}{4}}sech^2(mx)$$ $$\omega^2_1=3m^2$$ $$\psi_1(x)=\sqrt{\frac{3m}{2}}tanh(mx)sech(mx)$$

And the normalized and generalized eigenfunctions for the continious part of the spectra respectively

$$k\geq0$$ $$Y_k(x)=\frac{(tanh(mx)-ik)e^{ikx}}{\sqrt{k^2+m^2}}e^{i\delta_k}$$ $$k<0$$ $$Y_k(x)=\frac{(tanh(mx)-ik)e^{ikx}}{\sqrt{k^2+m^2}}e^{-i\delta_k}$$

and $$k\geq0$$

$$Y_k(x)=\frac{(-k^2-3imktanh(mx)+4m^2+6m^2sech^2(mx))e^{ikx}}{\sqrt{(k^2+m^2)(k^2+4m^2)}}e^{i\delta_k}$$

$$k<0$$

$$Y_k(x)=\frac{(-k^2-3imktanh(mx)+4m^2+6m^2sech^2(mx))e^{ikx}}{\sqrt{(k^2+m^2)(k^2+4m^2)}}e^{-i\delta_k}$$

where $$\delta_k$$ are the phase factors. These functions can easely be found under use of the ladder structure and an ansatz of plane waves. They form a set of generalized eigenfunction under construction of a gelfand triple. The second quantization for the vacuum sectors is now straight forward as in the free case, but the kink sector is a bit tricky in terms of fock space construction and introduction of a distorted fourier transform associated with the generalized eigenfunctions. It should be also clear, that the zero modes cause somee trouble regarding divergencies of the hamiltionian. They have to be eliminated by normal ordering with respect to the kink what is called kink representation and there exists a unitary map between kink and vacuum representation which involves the semiclassical 1-loop mass correction. Basicly the fields and the conjugated momenta in heisenberg representation associated to the SUSY-Chain look like this after a lot of struggle.

$$\phi(x,t)=-\sqrt{M_{cl}}\mathcal{X}\psi_0+\displaystyle\sum_{n=1}^{N-1} \frac{1}{\sqrt{\omega_n}}(a_n(t)+a^{\dagger}_n(t))\psi_n+\frac{1}{\sqrt{2\pi}}\int\mathrm {\frac{1}{\sqrt{\omega_k}}}(a_k(t)Y_k(x)+a^{\dagger}_k(t)\overline{Y_k(x)})$$

$$\pi(x,t)=-\frac{\mathcal{P}}{\sqrt{M_{cl}}}\psi_0-\displaystyle\sum_{n=1}^{N-1} \frac{i\sqrt{\omega_n}}{\sqrt{2}}(a_n(t)+a^{\dagger}_n(t))\psi_n+\frac{1}{\sqrt{2\pi}}\int\mathrm {\frac{-i\sqrt{\omega_k}}{\sqrt{2}}}(a_k(t)Y_k(x)-a^{\dagger}_k(t)\overline{Y_k(x)})$$

These satisfy the CCR for all schwartz functions and it can be shown that there exist fock spaces on which the associated hamiltonians to the systems are self adjoint operators. I have to read the paper of Coleman again but it should be possible in priniple to construct a dual fermion system to any of the elements of the chain like it was shown by coleman for Sin-Gordon in the 70s.

Edit: I have to slow down my enthusiasm a bit because there exists no multikink configuration in $$\phi^4$$ as it is present in Sin-Gordon. Maybe one should concentrate on the dilute kink gas in $$\phi^4$$ if someone searches for a fermion field configurations dual to the scalar field configurations. But that fermion model cannot be a multiparticle system if you want it to have the same duality like the massive thirring model and the sin gordon model have, because there are no solutions with topological charge greater than $$\pm 1$$ due to the vacuum structure and the charge is associated with the fermion number. If a duality exists it has to be of completely different nature.

• Could you clarify a little bit? Are you talking about the semi-classical approximation expanding about the soliton (which involves a Pöschl-Teller potential if I recall)? The semiclassical approximation is only approximate for $\phi^4$, and need not work well near the phase transition, which is where the description in terms of Majorana fermions is most useful. I also don't see where SUSY comes in, and I'm not sure what your $N$ is referring to. Mar 10, 2020 at 21:59
• Thank you for clarifying. It is as I expected, the SUSY refers to SUSY QM with the Pöschl-Teller potential. Notice that your fluctuation operator is only expanding to second order, but $\phi^4$ is not integrable and higher order corrections matter. The reason you can do so much with the semi-classics in sin-Gordon is because it is integrable. I don't think this answers the question but I upvoted in any case Mar 11, 2020 at 18:21
• Yes the absence of a breather in $\phi^4$ . Sad :/
As you know, the transverse field Ising model in 1D can be mapped to Majorana fermions, and the critical phase corresponds to a free Majorana. On the other hand we know that free Majorana can be bosonized to a $$(\nabla \phi)^2$$ (massless boson) theory. You already wrote the mapping from fermions to spins, so you then just need to find how exactly the spins get mapped to the boson theory. Sachdev's book on quantum phase transitions probably contains discussion of this.