Abstract
In the present work, the set of stationary solutions of the Gross-Neveu model in 't Hooft limit is extended. Such extension is obtained by striving a hidden supersymmetry associated to disconnected sets of stationary solutions. How the supersymmetry arises from the Darboux-Miura transformations between Lax pairs of the stationary modified Korteweg-de Vries and the stationary Korteweg-de Vries hierarchies is shown, associating the correspondent superpotentials to self-consistent condensates for the Gross-Neveu model.
Highlights
The Gross-Neveu (GN) model [1] corresponds to a quantum field theory for nonlinear interacting fermions without mass
Models with interacting fermions and self-consistent condensates have been used to describe a large variety of phenomena related to soliton physics, kinks, and breathers, especially in particle physics [2,3], superconductivity [4,5,6], and conducting polymer models [7,8,9], among other areas [10,11]
The first analytical solutions in this direction were obtained by applying the inverse scattering method [12,13], which allowed one to relate fermionic condensates to superpotentials of pairs of reflectionless systems of the Schrödinger type in 1 þ 1 dimension (1 þ 1D), bringing to light a hidden nonlinear N 1⁄4 4 supersymmetry in the stationary sector of the GN model [14,15,16]
Summary
The Gross-Neveu (GN) model [1] corresponds to a quantum field theory for nonlinear interacting fermions without mass. The self-consistency equation (2.3) corresponds to a system of equations that defines the occupation of each physical state of the spectra of HD by the different flavors Another important behavior of the Lax pair operators is a Burchnall-Chaundy type relationship between matrix differential operators, which relates powers of the Lax pair operators in the following form: 2Y n−1 PDn 2 1⁄4 Pn;BCðHDÞ 1⁄4 ðHD − ElÞ; ð2:13Þ which defines the eigenvalues zD of HD and yD of PD over a hyperelliptic curve. By observing intertwining operators that generate such transformations [A d dx þ vðxÞ], families of superpotentials vðxÞ will be found, which will be solutions of the s-mKdVh and candidates to self-consistent stationary condensates of the Gross-Neveu model.
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