Could this model have soliton solutions?

Question

We consider a theory described by the Lagrangian,

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

The corresponding field equations are, $$(i\gamma^\mu\partial_\mu-m+g\bar{\Psi}\Psi)\Psi=0$$

Could this model have soliton solutions? Without the last term, it is just a Dirac field (if $g=0$), but it has to be included. This 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?

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny

Comments

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.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Ron Maimon

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I guess m is positive as in dirac equation. What is the problem?

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny

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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).

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Ron Maimon

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Actually Im trying to know if some kind of soliton (or at least a solitary wave) is possible in this model.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny

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@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.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Ron Maimon

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I'm not sure if it is very important, but I'd want to know why you say that a classical solution is not going to work. I wonder if this model is known, as I say before I I haven't found it in any reference and I would like to know if you have any.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny

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@Anthony: Fermionic fields don't have classical solutions. This model is heavily studied--- it's the Gross Neveu model.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Ron Maimon

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@Anthony: The solitons are for the fermion bilinears, and the relevant literature is Witten's "Nonabelian bosonization" of the late 1970s, 1978 or thereabouts. You are giving the 1 component model, which is "abelian bosonization" because it's a U(1) current algebra. U(1) can have solitons in 2d because the U(1) can wrap in a large circle in 2d, but the solitons are bosonic, they are like the BCS condensate solitons, not Fermions directly. I read these papers ages ago, but never worked with it, I'll try to write a proper answer, but it needs a little thought.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Ron Maimon

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Thank you very much, I have never heard before that name, Gross-Neveu. If you can write an answer I will be very grateful.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny

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@Anthonny, I have noticed you have not accepted any answers from your physics colleagues. Go through some old answers and accept some, as some of them, if not most, look well answered! :]

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Killercam

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@RonMaimon: You should change your comment into an answer. We could vote it even if the OP does not close it.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Jon

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@Jon: I would like to review Witten's article before doing so. I only get a chance to go to the library once a week, If you have institutional access, this is the nonabelian bosonization current algebra stuff.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Ron Maimon

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But how do we know in general, just by looking at (any) Lagrangian, whether or not it will have soliton solutions? (sorry I joined late ... 2.5 yrs late ...)

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user New_new_newbie