The Approach to the Thermodynamic Limit in Lattice QCD at \mu\neq0

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 $\mu<m_\pi/2$ 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 \to \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 one. The results for a finite volume compare well with lattice results obtained by Allton {\it et al}.. After taking appropriate limits, we reproduce previously derived results for the $\epsilon$-regime and for 1-dimensional QCD. The distribution of the phase itself is also computed.