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QCD at finite quark-/baryon-number density, which describes nuclear matter,has a sign problem which prevents direct application of standard simulationmethods based on importance sampling. When such finite density is implementedby the introduction of a quark-number chemical potential $\mu$, this manifestsitself as a complex fermion determinant. We apply simulations using the ComplexLangevin Equation (CLE) which can be applied in such cases. However, this isnot guaranteed to give correct results, so that extensive tests are required.In addition, gauge cooling is required to prevent runaway behaviour. We testthese 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 smallquark 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 transitionat $\mu \approx m_N/3$ and a saturation density of $3$ at large $\mu$, theobservables show departures from known values at small $\mu$. We are nowrunning on a larger lattice ($16^4$) at weaker coupling $\beta=5.7$. At $\mu=0$this significantly improves agreement between measured observables and knownvalues, and there is some indication that this continues to small $\mu$s. Thisleads one to hope that the CLE might produce correct results in theweak-coupling -- continuum -- limit.

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