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

TL;DR

This paper explores the crystalline phase of the Gross-Neveu model with a chemical potential using three methods, revealing inhomogeneous phases, new scales, and nonperturbative effects consistent across approaches.

Contribution

It provides a comprehensive analysis of the crystalline phase in the Gross-Neveu model via perturbative, semiclassical, and integrability methods, detailing nonperturbative scales and effects.

Findings

Inhomogeneous phase emerges at high chemical potential.

Two new nonperturbative scales, Λ_n and Λ_c, govern the physics.

Oscillatory chiral condensate profile characterized at large N.

Abstract

We study the crystalline phase of the $O(2N)$ Gross--Neveu model with a chemical potential for $a \leq N-2$ 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 $\Lambda_{\rm n}$ and $\Lambda_{\rm c}$. 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.