Finite size effects in the Gross-Neveu model with isospin chemical potential

TL;DR

This paper investigates finite size effects on the phase structure of the two-flavored Gross-Neveu model with isospin chemical potential in (1+1) dimensions, revealing how boundary conditions and system size influence pion condensation.

Contribution

It provides a detailed analysis of how finite spatial size and boundary conditions affect pion condensation phases in the Gross-Neveu model with isospin chemical potential.

Findings

Pion condensation occurs at any nonzero isospin chemical potential when the system size is infinite.

Finite size introduces a phase diagram in the parameters  and 1/L, with a critical size .

In the pion condensed phase, the gap oscillates with system size and chemical potential.

Abstract

The properties of the two-flavored Gross-Neveu model in the (1+1)-dimensional $R^1\times S^1$ spacetime with compactified space coordinate are investigated in the presence of the isospin chemical potential $\mu_I$. The consideration is performed in the limit $N_c\to\infty$, i.e. in the case with infinite number of colored quarks. It is shown that at $L=\infty$ ($L$ is the length of the circumference $S^1$) the pion condensation phase is realized for arbitrary small nonzero $\mu_I$. At finite values of $L$, the phase portraits of the model in terms of parameters $\nu\sim\mu_I$ and $\lambda\sim 1/L$ are obtained both for periodic and antiperiodic boundary conditions of the quark field. It turns out that in the plane $(\lambda,\nu)$ there is a strip $0\le\lambda<\lambda_c$ which lies as a whole inside the pion condensed phase. In this phase the pion condensation gap is an oscillating function vs both $\lambda$ (at fixed $\nu$) and $\nu$ (at fixed $\lambda$).