Approach to the thermodynamic limit in lattice QCD atμ≠0

Abstract

The expectation value of the complex phase factor of the fermion determinant is computed to leading order in the $p$ expansion of the chiral Lagrangian. The computation is valid for $\ensuremath{\mu}<{m}_{\ensuremath{\pi}}/2$ and determines the dependence of the sign problem on the volume and on the geometric shape of the volume. In the thermodynamic limit with ${L}_{i}\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed temperature $1/{L}_{0}$, the average phase factor vanishes. In the low temperature limit where ${L}_{i}/{L}_{0}$ is fixed as ${L}_{i}$ becomes large, the average phase factor approaches 1 for $\ensuremath{\mu}<{m}_{\ensuremath{\pi}}/2$. The results for a finite volume compare well with lattice results obtained by Allton et al. After taking appropriate limits, we reproduce previously derived results for the $ϵ$ regime and for one-dimensional QCD. The distribution of the phase itself is also computed.