Perturbative, Nonperturbative and Exact Aspects of Crystalline Phases in the Gross-Neveu Model

Abstract

We study the crystalline phase of the $O(2N)$ Gross--Neveu model with a chemical potential for $a\le N-2$ of the fermions. We analyze the problem in three independent ways: using perturbative QFT methods, a semiclassical large $N$ analysis, and integrability techniques (both at finite and large $N$). The resulting picture is consistent across all three approaches: at sufficiently large chemical potential $h$, an inhomogeneous phase emerges in which $a$-particle bound states condense and which, at large $N$, corresponds to a periodically oscillating chiral condensate. In this phase, the usual dynamically generated scale $\Lambda$ is replaced by two new dynamically generated scales ${\Lambda }_{\mathrm{n}}$ and ${\Lambda }_{\mathrm{c}}$. These two scales govern the multiple nonperturbative effects in the theory, corresponding in particular to the mass gaps of neutral and charged excitations on top of the inhomogeneous vacuum, respectively. They also control the nonperturbative corrections to observables such as the free energy and provide the parameters characterizing the oscillatory profile of the mean field at large $N$. In this paper, we provide the necessary details of each of the three methods, thereby complementing the results announced in a previous, shorter publication.

Comments: 61 pages + 27 pages appendices, 18 figures