Gross-Neveu Models, Nonlinear Dirac Equations, Surfaces and Strings

Abstract

Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model (GN2), and its chiral cousin, the NJL2 model, have shown that there are phases with inhomogeneous crystalline condensates. These (static) condensates can be found analytically because the relevant Hartree-Fock and gap equations can be reduced to the nonlinear Schrödinger equation, whose deformations are governed by the mKdV and AKNS integrable hierarchies, respectively. Recently, Thies et al have shown that time-dependent Hartree-Fock solutions describing baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation, and can be mapped directly to classical string solutions in AdS3. Here we propose a geometric perspective for this result, based on the generalized Weierstrass spinor representation for the embedding of 2d surfaces into 3d spaces, which explains why these well-known integrable systems underlie these various Gross-Neveu gap equations, and why there should be a connection to classical string theory solutions. This geometric viewpoint may be useful for higher dimensional models, where the relevant integrable hierarchies include the Davey-Stewartson and Novikov-Veselov systems.

27 pages, 1 figure

References in corpus (15)

• Quarkyonic Chiral Spirals
• Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space
• Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems
• Pohlmeyer reduction of AdS_5 x S^5 superstring sigma model
• How many phases meet at the chiral critical point?
• All about the Static Fermion Bags in the Gross-Neveu Model
• Stringy NJL and Gross-Neveu models at finite density and temperature
• The Bogoliubov/de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry
• Generalized Weierstrass representation for surfaces in multidimensional Riemann spaces
• The Gross-Neveu Model from String Theory
• Surfaces in the four-space and the Davey--Stewartson equations
• Deformation surfaces, integrable systems and Chern - Simons theory
• Dimensional Reduction of Fermions in Brane Worlds of the Gross-Neveu Model
• The U(1) Gross-Neveu Model at Non-Zero Chemical Potential
• Four Fermion Models at Non-Zero Density

Cited by in corpus (13)

• Interweaving Chiral Spirals
• Inhomogeneous charged pion condensation phenomenon in the NJL${}_2$ model with quark number and isospin chemical potentials
• Chiral density waves in the NJL${}_2$ model with quark number and isospin chemical potentials
• Exotic supersymmetry of the kink-antikink crystal, and the infinite period limit
• Transmutations of supersymmetry through soliton scattering, and self-consistent condensates
• The nonlinear Dirac equation in Bose-Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices
• Fermion in a multi-kink-antikink soliton background, and exotic supersymmetry
• Baryon-baryon scattering in the Gross-Neveu model: the large N solution
• Evidence for factorized scattering of composite states in the Gross-Neveu model
• Quantum quench, large N, and symmetry restoration
• Exploring new states of matter with a photonic emulator
• Supersymmetric construction of self-consistent condensates in large N GN model: solitons on finite-gap potentials
• SU(N) affine Toda solitons and breathers from transparent Dirac potentials