# Could this model have soliton solutions?

$\mathcal{L}=i\bar{\Psi}\gamma^\mu\partial_\mu\Psi-m\bar{\Psi}\Psi+\frac{1}{2}g(\bar{\Psi}\Psi)^2$

Field equation $(i\gamma^\mu\partial_\mu-m+g\bar{\Psi}\Psi)\Psi=0$

Could this model have soliton solutions?

If you have realized it is a Dirac field if g=0. I have added the last term. It is similar to the Thirring model. I was looking for this field in books and papers but I haven't found it. If you know about it could you give me any reference?

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I guess you are trying to make a Fermionic mexican hat. Please say so--- because either sign of m in the action gives a positive mass for the Fermion. – Ron Maimon Oct 24 '11 at 19:03
I guess m is positive as in dirac equation. What is the problem? – Anthonny Oct 24 '11 at 19:19
Oh--- ok--- I was wrong. I thought you were trying to make solitons like those that occur in the bosonic form of this action (which doesn't work). – Ron Maimon Oct 24 '11 at 19:25
@Anthony: this model is Fermionic. Solitons are coherent superpositions bosonic excitations. But the model conserves a U(1) charge which counts the Fermions, so that you can make a Fermi sea with a large numbers of fermions, and perhaps get a superconducting condensate, which can then have solitons. But I don't think this is what you meant. Perhaps you can say exactly what kind of soliton you are after? If you want a classical solution of the form $\psi(x)$, it's not going to work, because $\psi$ is Fermi. – Ron Maimon Oct 24 '11 at 23:00
@Anthony: Fermionic fields don't have classical solutions. This model is heavily studied--- it's the Gross Neveu model. – Ron Maimon Oct 25 '11 at 15:10