Solving the Gross-Neveu model with relativistic many-body methods

Abstract

The Gross-Neveu model provides a unique opportunity to apply relativistic many-body techniques (Dirac-Hartree approximation, RPA) in a context where all calculations can be done analytically and -in the large N limit -yield the exact results. The physical fermion as well as multifermion ("baryon") and fermion-antifermion ("meson") bound states are discussed in this spirit, with special emphasis on the role of the Dirac sea.

No first-order phase transition in the Gross-Neveu model?

“…Using units where m = 1 (m is the physical fermion mass), it is given by [18,19] S(x) = 1 + y(tanh ξ − − tanh ξ + ) , ξ ± = yx ± 1 2 artanh y .…”

“…The k n are the single particle momenta to the corresponding discrete p n , obtained via the transcendental equation (19).…”

Kink dynamics, sinh-Gordon solitons and strings in AdS3from the Gross–Neveu model

“…Known type II solutions are the kink-antikink baryons [3,11], the kink crystal at finite temperature [13] and the time-dependent breather [3] to be discussed below in more detail. Incidentally, all HF solutions of the massive GN model (i.e., Lagrangian (1) supplemented by a term −m 0 N k=1ψ k ψ k ) are also of type II [14][15][16], and no solution of type III or higher is known in the GN model which would require more than two functions and hence more than two space-time independent self-consistency conditions.…”

Analytical solution of the Gross-Neveu model at finite density

“…The coupling constant is related to the cutoff via the (vacuum) gap equation [2,9] which reads (in units where the vacuum fermion mass is 1)…”

“…We therefore conclude that the true ground state happens to lie on the oneparameter trajectory of potentials which can be dealt with analytically. In order to show this without invoking any numerical results, one still has to verify that the ground state expectation value ofψψ is self-consistent, as was done for the single baryon in [9]. This is indeed possible, but requires more details about the rather involved Lamé wave functions [5] as well as some patience in juggling identities for elliptic functions.…”