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

Abstract

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