Abstract
We consider a version of the Gross-Neveu model in 1+1 dimensions with discrete chiral and continuous flavor symmetry (isospin). In 2+1 dimensions, this model is known as chiral Heisenberg Gross-Neveu model. Spontaneous symmetry breaking and the emergence of two massless and one massive scalar bosons are shown. A duality to the Nambu--Jona-Lasinio model with isospin is exhibited, provided that the isovector pseudoscalar mean field is constrained to a plane in isospin space. This enables us to find the phase diagram as a function of temperature, chemical potential and isospin chemical potential as well as twisted kinks. A bare mass term acts quite differently when added to this model as compared to other chiral variants of the Gross-Neveu model.
Highlights
Four-fermion models in 1 þ 1 dimensions can teach us a lot about strongly interacting relativistic systems
As a matter of fact, in 2 þ 1 dimensions this model is known in the condensed matter literature as the “chiral HeisenbergGross-Neveu model” [5], presumably because the interaction term is reminiscent of the spin-spin interaction in the Heisenberg model of magnetism
Questions which immediately come to one’s mind are the following: Does the model possess twisted kinks like the other GN variants, and how can one find them? What does the phase diagram look like, notably regarding inhomogeneous phases? Can one find explicit time dependent scattering solutions, and what can be said about the integrability of the model? We shall see that it takes little more than a duality to infer many physical properties of the isoGN model from previous results for the isoNJL model
Summary
Four-fermion models in 1 þ 1 dimensions can teach us a lot about strongly interacting relativistic systems. Studies have uncovered a rich variety of fermionantifermion and multifermion bound states, time dependent scattering problems, as well as nontrivial phase diagrams as a function of temperature and chemical potentials. This brief survey suggests to add one more variant to this list, which seems to have been forgotten so far. As a matter of fact, in 2 þ 1 dimensions this model is known in the condensed matter literature as the “chiral HeisenbergGross-Neveu model” [5], presumably because the interaction term is reminiscent of the spin-spin interaction in the Heisenberg model of magnetism. We shall see that it takes little more than a duality to infer many physical properties of the isoGN model from previous results for the isoNJL model.
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