Analytical solution of the Gross-Neveu model at finite density

Abstract

Recent numerical calculations have shown that the ground state of the Gross-Neveu model at finite density is a crystal. Guided by these results, we can now present the analytical solution to this problem in terms of elliptic functions. The scalar potential is the superpotential of the nonrelativistic Lamé Hamiltonian. This model can also serve as analytically solvable toy model for a relativistic superconductor in the Larkin-Ovchinnikov-Fulde-Ferrell phase.In this paper we reconsider the simplest variant of the Gross-Neveu (GN) model, a 1+1 dimensional relativistic field theory with N species of fermions interacting via a quartic self-interaction [1]

Self-consistent large-N analytical solutions of inhomogeneous condensates in quantum ℂPN − 1 model

“…Self-consistent analytical solutions such as a real kink [53,56], a twisted (complex) kink [54], a real kink-anti-kink (polaron) [53,63,64], a real kink-anti-kink-kink [55,65,66] and more general real solutions [67] have been known. Recently, a theoretical progress has been achieved for inhomogeneous condensates in the 1+1 dimensional (chiral) GN model, e.g., the exact self-consistent and inhomogeneous condensates such as a real kink crystal [68,69] (Larkin-Ovchinnikov(LO) state [70]), a chiral spiral (Fulde-Ferrell(FF) state [71]), and a twisted kink crystal [72,73] (FF-LO state) have been found by mapping the equations to the nonlinear Schrödinger equation, and such states have been shown to be ground states in a certain region of the phase diagram for finite temperature and density [74]. More generally, multiple twisted kinks with arbitrary phase and positions [75,76] can be further constructed systematically due to the integrable structure behind the model known as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy for the nonlinear Schrödinger equation [77][78][79].…”

Chiral spirals from noncontinuous chiral symmetry: The Gross-Neveu model results

“…For the description of such phases, the model studies rely on the understanding of their twodimensional counterparts: the Gross-Neveu (GN) model [7,8] as a counterpart for the NJL 4 model [3], the 't Hooft model (QCD 2 ) [9] for the confining model [5,6], and the NJL 2 model [10] for the extended NJL 4 model with tensor 4-Fermi interactions [11]. In fact, the solutions of twodimensional models can be naturally embedded into the four-dimensional mean field ansatz.…”

“…In this paper we explain how to understand differences between them by revisiting inhomogeneous solutions of the GN model [7]. To avoid confusion, we emphasize that we will not attempt to modify the analytic solution, which was shown to achieve the ground state [8].…”

Interacting fermions, boundaries, and finite size effects

“…[2], is extraordinarily rich and many of its properties were understood only comparatively recently (see, for example, Refs. [3][4][5][6]). A very nice review of its phase diagram can be found in Ref.…”

“…Substantial efforts were carried out to include such spatially varying phases, and the interested reader may consult Refs. [3][4][5][6] for details and additional bibliography. In the following, we will restrict our analysis to the case of spatially constant phases and leave the inhomogeneous case for future work.…”