Complex Langevin for Lattice QCD at $T=0$ and $\mu \ge 0$

Abstract

QCD at finite quark-/baryon-number density, which describes nuclear matter, has a sign problem which prevents direct application of standard simulation methods based on importance sampling. When such finite density is implemented by the introduction of a quark-number chemical potential $\mu$, this manifests itself as a complex fermion determinant. We apply simulations using the Complex Langevin Equation (CLE) which can be applied in such cases. However, this is not guaranteed to give correct results, so that extensive tests are required. In addition, gauge cooling is required to prevent runaway behaviour. We test these methods on 2-flavour lattice QCD at zero temperature on a small ($12^4$) lattice at an intermediate coupling $\beta=6/g^2=5.6$ and relatively small quark mass $m=0.025$, over a range of $\mu$ values from $0$ to saturation. While this appears to show the correct phase structure with a phase transition at $\mu \approx m_N/3$ and a saturation density of $3$ at large $\mu$, the observables show departures from known values at small $\mu$. We are now running on a larger lattice ($16^4$) at weaker coupling $\beta=5.7$. At $\mu=0$ this significantly improves agreement between measured observables and known values, and there is some indication that this continues to small $\mu$s. This leads one to hope that the CLE might produce correct results in the weak-coupling -- continuum -- limit.