All about the Static Fermion Bags in the Gross-Neveu Model

TL;DR

This paper comprehensively reviews all stable static fermion bag solutions in the 1+1 dimensional Gross-Neveu model, introducing new topological solitons, and provides detailed derivations and pedagogical appendices on related mathematical tools.

Contribution

It identifies and characterizes all stable static fermion bags, including new topological solitons, with explicit formulas and detailed analysis, expanding understanding of the model's soliton spectrum.

Findings

Discovery of heavier topological solitons (HTS) as bound states of kinks and trivial solitons.

Mass of HTS equals sum of constituent masses, independent of separation distance.

No additional stable static solitons beyond those identified.

Abstract

We review in detail the construction of {\em all} stable static fermion bags in the 1+1 dimensional Gross-Neveu model with $N$ flavors of Dirac fermions, in the large $N$ limit. In addition to the well known kink and topologically trivial solitons (which correspond, respectively, to the spinor and antisymmetric tensor representations of O(2N)), there are also threshold bound states of a kink and a topologically trivial soliton: the heavier topological solitons (HTS). The mass of any of these newly discovered HTS's is the sum of masses of its solitonic constituents, and it corresponds to the tensor product of their O(2N) representations. Thus, it is marginally stable (at least in the large $N$ limit). Furthermore, its mass is independent of the distance between the centers of its constituents, which serves as a flat collective coordinate, or a modulus. There are no additional stable static solitons in the Gross-Neveu model. We provide detailed derivation of the profiles, masses and fermion number contents of these static solitons. For pedagogical clarity, and in order for this paper to be self-contained, we also included detailed appendices on supersymmetric quantum mechanics and on reflectionless potentials in one spatial dimension, which are intimately related with the theory of static fermion bags. In particular, we present a novel simple explicit formula for the diagonal resolvent of a reflectionless Schr\"odinger operator with an arbitrary number of bound states. In additional appendices we summarize the relevant group representation theoretic facts, and also provide a simple calculation of the mass of the kinks.